Tangent Weights and Invariant Curves in Type A Bow Varieties
Alex Foster & Yiyan Shou

§ INTRODUCTION

For a broad class of torus-equivariant complex algebraic varieties $T\, \rotatebox[origin=c]{-90}{$\circlearrowright$}\, Y$, the fixed points and invariant curves under the torus action play an important role in the study of the equivariant cohomology of $Y$. Here, we define an invariant curve to be the closure of a 1-dimensional $T$-orbit. This paper extends the work of [8] on the case where $Y$ is a Cherkis bow variety of type A by providing combinatorial codes for invariant curves.

For simplicity, assume that $Y$ has finitely many fixed points. Under certain conditions, the map $\iota^*:H^*_T (Y) \to H^*_T (Y^T)$ induced by the inclusion of the fixed point locus is injective. In other words, each equivariant cohomology class is uniquely determined by its fixed point restrictions. This is the case for partial flag varieties, and more generally, Nakajima quiver varieties of type A. As $H^*_T(\pt) = \CC[u_1, ..., u_m]$, where $m = \dim(T)$, this allows us to describe equivariant cohomology classes as tuples of polynomials. This approach is taken in <cit.> to study $\hbar$-deformed Schubert classes. In the setting where the variety is equivariantly formal with respect to the $T$-action, we can describe the image of $\iota^*$ in terms of the fixed points and invariant curves. For instance, when $Y$ possesses finitely many invariant curves, the result of [4] describes the image of $\iota^*$ in terms of simple matching conditions on the polynomials (see also <cit.>). These matching conditions are determined by the invariant curves and their tangent data. When there are infinitely many invariant curves, the Chang-Skjelbred lemma [3] provides a more complicated description (see also [2]). Unfortunately, not every Cherkis bow variety is equivariantly formal nor has injective restriction map <cit.>. The appendix of <cit.> shows, however, that there is a natural subalgebra (generated by the stable envelope) of $H^*_T(Y)$ on which $\iota^*$ is injective. Determining whether this subalgebra can be described in terms of fixed points and invariant curves is an open problem.

Besides providing a concrete way to describe equivariant cohomology classes, the fixed points and invariant curves play an important role in the study of stable envelopes <cit.>. The stable envelope is an axiomatically defined $T$-equivariant characteristic class with important connections to quantum integrable systems, quantum groups, and quantum cohomology. In certain settings, the axioms can be formulated as conditions on fixed point restrictions that are entirely determined by the invariant curves and their tangent data. This formulation is, in a sense, local to the fixed points, in contrast to the original global axioms of <cit.>. See <cit.> for a discussion of local stable envelope axioms in the Schubert calculus setting and <cit.> for the Cherkis bow variety setting. In the Schubert calculus setting, the stable envelope agrees with the Chern-Schwartz-MacPherson classes of Schubert cells <cit.>. While we are focusing on equivariant cohomology here, this discussion generalizes to K-theory and elliptic cohomology. In the elliptic cohomology setting, the stable envelope plays an important role in the study of 3d ($N = 4$) mirror symmetry <cit.>.

The work of <cit.> shows that Cherkis bow varieties <cit.> form a natural pool of varieties where the phenomenon of mirror symmetry manifests itself in the combinatorics and elliptic stable envelope. Bow varieties are torus-equivariant holomorphic symplectic manifolds that generalize Nakajima quiver varieties <cit.>. In [8], a combinatorial framework for studying the fixed points of bow varieties of type A is established. This paper completes the picture by providing a complete combinatorial description of the invariant curves. A construction for compact invariant curves involving so called “butterfly surgeries" was previously known to the second author—see [10]. However, whether that construction captured all compact invariant curves remained a conjecture. This work resolves that conjecture in the affirmative and extends the result by also capturing the noncompact invariant curves using modified “butterfly surgery" constructions.

While the main accomplishment of this paper is the complete classification of invariant curves, several intermediate developments are required, some of which are interesting in their own right. In Section <ref>, we review the combinatorial framework of [8] for the study of type A Cherkis bow varieties. In Section <ref>, we prove a novel formula for the tangent weights at a fixed point of a bow variety. This formula can be used to efficiently calculate tangent weights by hand or by computer and plays a pivotal role in the proof of our main result. In Section <ref>, we review the essentials of torus invariant curves and give a self-contained presentation of the butterfly surgery construction of [10]. This section also describes two novel constructions for noncompact invariant curves. Finally, we reformulate these constructions in terms of surgery operations on Young diagrams, resulting in a description of invariant curves in terms of familiar combinatorics. Section <ref> further reformulates the combinatorics of Section <ref> in terms of binary contingency tables (BCTs). The results of this section provide a simple algorithm for constructing all invariant curves in a bow variety. Section <ref> contains the main result, a classification of invariant curves containing a given fixed point. Curves are characterized as one of three types, each having its own unique properties. In the final section, Section <ref>, we apply the machinery developed in the previous sections to the example bow varieties of [8].

§ BACKGROUND

We begin by recalling the essential pieces of the combinatorial framework of [8] for the study of bow varieties. We will require brane diagrams, tie diagrams, table-with-margins, butterfly diagrams, the formula for the tangent bundle of a bow variety, formulas for fixed point restrictions, and the Hanany-Witten transition.

§.§ Brane Diagrams and Bow Varieties

The [8] framework is based on the [7] construction of bow varieties. Here, we provide a simplified presentation. A brane diagram $\brane$ consists of a horizontal line divided into segments by $n$ lines of slope 1 and $m$ lines of slope $-1$. The lines must be arranged so as not to create a triple (or greater) intersection. Additionally, each segment is decorated by a nonnegative integer multiplicity where the infinite segments on the far left and right are automatically decorated with 0. An example of a brane diagram is shown in Figure <ref>. The lines of slope 1 are called “NS5 branes”, and we label them from left to right as $V_1, ..., V_n$. The lines of slope $-1$ are called “D5 branes”, and we label them from left to right as $U_1, ..., U_m$. The NS5 and D5 branes are referred to collectively as “5-branes". The segments along with their multiplicities are referred to as “D3 branes". The segments are labeled as $X_0, ..., X_{n + m}$ and the multiplicity of a D3 brane $X$ is denoted by $d_X$. In particular, $d_{X_0} = d_{X_{n + m}} = 0$.

[thick,red] (0.5,0) –(1.5,2);
[red] at (.5,-.6) $V_1$;
[thick] (1.5,1)–(2.5,1) node [above] $2$ – (3.5,1);
[thick,blue](4.5,0) –(3.5,2);
[thick](4.5,1)–(5.5,1) node [above] $2$ – (6.5,1);
[thick,red](6.5,0) – (7.5,2);
[red] at (6.5,-.6) $V_2$;
[thick](7.5,1) –(8.5,1) node [above] $2$ – (9.5,1);
[thick,blue] (10.5,0) – (9.5,2);
[thick] (10.5,1) –(11.5,1) node [above] $4$ – (12.5,1);
[thick,red](12.5,0) – (13.5,2);
[thick](13.5,1) –(14.5,1) node [above] $3$ – (15.5,1);
[thick,red] (15.5,0) – (16.5,2);
[thick](16.5,1) –(17.5,1) node [above] $3$ – (18.5,1);
[thick,red](18.5,0) – (19.5,2);
[thick](19.5,1) –(20.5,1) node [above] $4$ – (21.5,1);
[thick,blue](22.5,0) – (21.5,2);
[blue] at (21.5,2.4) $U_3$;
[thick](22.5,1) –(23.5,1) node [above] $3$ – (24.5,1);
[thick,red] (24.5,0) – (25.5,2);
[blue] at (27.5,2.4) $U_4$;
[blue] at (30.5,2.4) $U_5$;
[thick] (25.5,1) –(26.5,1) node [above] $2$ – (27.5,1);
[thick,blue](28.5,0) – (27.5,2);
[thick](28.5,1) –(29.5,1) node [above] $2$ – (30.5,1);
[thick,blue](31.5,0) – (30.5,2);

[black,dashed,->](7,-5) to [out=170,in=270] (0.3,-1);
[black,dashed,->](7,-5) to [out=110,in=270] (6.3,-1);
[black,dashed,->](7,-5) to [out=80,in=210] (12.3,-.3);
[black,dashed,->](7,-5) to [out=70,in=210] (15.3,-.3);
[black,dashed,->](7,-5) to [out=40,in=210] (18.3,-.3);
[black,dashed,->](7,-5) to [out=20,in=220] (24.3,-.3);
at (7,-5.5) NS5 branes;

[black,dashed,->](23,6) to [out=190,in=60] (3.7,2.25);
[black,dashed,->](23,6) to [out=230,in=80] (9.6,2.25);
[black,dashed,->](23,6) to [out=250,in=90] (21.7,3.25);
[black,dashed,->](23,6) to [out=270,in=100] (27.25,3.25);
[black,dashed,->](23,6) to [out=280,in=110] (30,3.25);
at (25.3, 6) D5 branes;

at (3,6) D3 branes;
[black,dashed,->](3,5) to [out=-90,in=90] (2,1.3);
[black,dashed,->](3,5) to [out=-80,in=90] (5,1.3);
[black,dashed,->](3,5) to [out=-70,in=100] (8,1.3);
[black,dashed,->](3,5) to [out=-65,in=110] (11,1.3);
[color=black] at (3.7,4.7) $\ldots$;

The first example of a brane diagram given in [8].

We do not bother displaying $X_0$ and $X_{n + m}$ in graphical representations of $\brane$. We also color the NS5 branes blue and the D5 branes red. These colors are purely for visual clarity and have no mathematical meaning.

To each brane diagram $\brane$, we associate a smooth, torus equivariant, symplectic holomorphic variety $\torus\, \rotatebox[origin=c]{-90}{$\circlearrowright$}\,\bowvar(\brane)$. This variety will be constructed as a GIT quotient. Given any brane $\beta$ (D5, NS5, or D3), denote the brane immediately to the left by $\beta^-$ and right by $\beta^+$.

First, a vector space will be associated to each 5-brane. For each D3 brane $X$, let $W_X = \CC^{d_X}$, and for each D5 brane $U$, let $\CC_U = \CC$.

* To each D5 brane $U$, associate a “triangle part"

\begin{align*}
\MM_U= &\Hom(W_{U^+},W_{U^-}) \oplus
\Hom(W_{U^+},\CC_U) \oplus \Hom(\CC_U,W_{U^-}) \\
&  \oplus\End(W_{U^-}) \oplus \End(W_{U^+}),
\end{align*}

whose elements will  be denoted by $(A_U, B_U,B'_U, a_U, b_U)$, as shown in the diagram

\[
\begin{tikzcd}
W_{U^-} \arrow[loop,out=200,in=160,distance=2em, "B_U", "\circ" marking] & & W_{U^+} \arrow[ld, "\circ" marking, "b_U" ] \arrow[ll, "A_U"] \arrow[loop,out=20,in=-20,distance=2em, "B'_U", "\circ" marking] \\
& \CC_U.\arrow[ul, "a_U"] &
\end{tikzcd}
\]

* To each NS5 brane $V$, associate the “two-way part"

\[
\MM_V= \Hom(W_{V^+},W_{V^-}) \oplus
\Hom(W_{V^-},W_{V^+}),
\]

whose elements will  be denoted by $(C_V, D_V)$, as shown in the diagram

\[
\begin{tikzcd}
W_{V^-} \ar[rr,bend right, "D_V"] & & W_{V^+}. \ar[ll, "C_V", bend right, "\circ" marking]
\end{tikzcd}
\]

Finally, define

\[
\MM=\bigoplus_{U\text{ D5}}\MM_U \oplus \bigoplus_{V\text{ NS5}}\MM_V,
\]

the sum of all triangle and two-way parts.

Next, define the groups

\[
\torus = \CC_\hbar^\times \times \prod_{U \text{ D5}}\CC_U^\times \quad\text{and}\quad \G = \prod_{X \text{ D3}} \mathrm{GL}(W_X),
\]

where $\CC_\hbar^\times$ and $\CC^\times_U$ are copies of $\CC^\times$. The group $\G$ acts on each $W_X$ and hence on $\MM$ in a natural way. The torus $\torus$ acts on $\MM$ by scaling $\CC_U$ by the $\CC_U^\times$ factor and scaling each circled map in the above diagrams by the $\CC^\times_\hbar$ factor.

Let $\MM_0$ be the subset of $\MM$ satisfying the following conditions:

* For each D5 brane $U$, we have $B_UA_U - A_UB'_U + a_Ub_U = 0$.

* For each D3 brane $X$, we have

* $B'_{X^-} - B_{X^+} = 0$ if $X$ is in between two D5 branes ($\U-X-\U$),

* $C_{X^+}D_{X^+} - D_{X^-}C_{X^-} = 0$ if $X$ is in between two NS5 branes ($\V-X-\V$),

* $-D_{X^-}C_{X^-} - B_{X^+} = 0$ if $X^-$ is an NS5 brane and $X^+$ is a D5 brane ($\V-X-\U$),

* $C_{X^+}D_{X^+} + B'_{X^-} = 0$ if $X^-$ is a D5 brane and $X^+$ is an NS5 brane ($\U-X-\V$).

These conditions arise from setting a certain moment map equal to 0 (see [7] or [8] for details). Hence, we will refer to them as “0-momentum" conditions. Let $\W = \bigoplus_{X\text{ D3}} W_X$, and define three additional “stability" conditions:

* For all D5 branes $U$, the only $B_{U^+}$-invariant subspace $S\subset W_{U^+}$ with $A_U(S) = 0, b_U(S) = 0$ is $S=0$.

* For all D5 branes $U$, the only $B_{U^-}$-invariant subspace $S\subset W_{U^-}$ with $\im(A_U)+\im(a_U) \subset S$ is $S=W_{U^-}$.

* The only subspace $S=\bigoplus_X S_X\subset \W$ invariant under all $A,B,C,D$ maps such that $\im(a_U) \subset S$ and $A_U$ induces an isomorphism $W_{U^+}/S_{U^+}\to W_{U^-}/S_{U^-}$ for all D5 branes $U$ is $S=\W$.

Stability conditions 1 and 2 are the S1 and S2 conditions of <cit.>, and stability condition 3 is the $\nu$2 condition of <cit.>. Let $\prequot$ be the subset of $\MM_0$ satisfying the three stability conditions. $\prequot$ is $\G$- and $\torus$-invariant.

Define the “Cherkis bow variety" corresponding to brane diagram $\brane$ as the quotient

\[
\bowvar(\brane) = \prequot / \G.
\]

There is a residual action of $\torus$ on $\bowvar(\brane)$, along with a collection of $\torus$-equivariant “tautological" vector bundles $\xi_X \to \bowvar(\brane)$, associated with the $W_X$ vector spaces. The rank of $\xi_X$ is $d_X$. A formula for the $\torus$-equivariant K-theory class of the tangent bundle $T\bowvar(\brane)$ is given in <cit.>:

* Let $\hb$ be the trivial line bundle whose fibres are scaled by the $\CC_\hbar^\times$ factor of $\torus$.

* For a D5 brane $U$ define

\begin{align*}
T_U =&
\Hom(\xi_{U^+},\xi_{U^-})\oplus
\hb\Hom(\xi_{U^+},\CC_U) \oplus
\Hom(\CC_U,\xi_{U^-})
\\
& \oplus \hb \End(\xi_{U^-})\oplus \hb\End(\xi_{U^+}).
\end{align*}

* For an NS5 brane $V$ define
T_V =\hb\Hom(\xi_{V^+},\xi_{V^-}) \oplus \Hom(\xi_{V^-},\xi_{V^+})

Then, as elements of $K^0_\torus (\bowvar(\brane))$, we have

\begin{equation}
\label{eqn:tangent_bundle}
\begin{aligned}
\left(\bigoplus_{U \text{ D5}} T_U \right)\oplus
\left( \bigoplus_{V \text{ NS5}} T_V \right) \\
\ominus \left( \bigoplus_{U \text{ D5}} \hb \Hom(\xi_{U^+},\xi_{U^-})  \right)
\ominus \left(\bigoplus_{X \text{ D3}} (1+\hb)\End(\xi_{X}) \right).
\end{aligned}
\end{equation}

§.§ Torus Fixed Points

[8] provides three combinatorial codes for the $\torus$-fixed points of a bow variety $\bowvar(\brane)$: tie diagrams, butterfly diagrams, and table-with-margins (BCTs). We will be utilizing all three in what follows.

Let us begin with tie diagrams. A “tie diagram" is obtained from a brane diagram by adding dashed lines, called “ties", between the 5-branes such that

* each tie joins 5-branes of different type (one NS5 and one D5 brane),

* for each segment $X$, there are $d_X$ distinct ties covering $X$.

In Figure <ref>, we added ties to the brane diagram in Figure <ref> to create a tie diagram. This tie diagram corresponds to one of the $\torus$-fixed points.

[thick,red] (0.5,0) –(1.5,2);
[thick] (1,1)–(2.5,1) node [above] $2$ – (31,1);
[thick,blue](4.5,0) –(3.5,2);
[thick](4.5,1)–(5.5,1) node [above] $2$ – (6.5,1);
[thick,red](6.5,0) – (7.5,2);
[thick](7.5,1) –(8.5,1) node [above] $2$ – (9.5,1);
[thick,blue] (10.5,0) – (9.5,2);
[thick] (10.5,1) –(11.5,1) node [above] $4$ – (12.5,1);
[thick,red](12.5,0) – (13.5,2);
[thick](13.5,1) –(14.5,1) node [above] $3$ – (15.5,1);
[thick,red] (15.5,0) – (16.5,2);
[thick](16.5,1) –(17.5,1) node [above] $3$ – (18.5,1);
[thick,red](18.5,0) – (19.5,2);
[thick](19.5,1) –(20.5,1) node [above] $4$ – (21.5,1);
[thick,blue](22.5,0) – (21.5,2);
[thick](22.5,1) –(23.5,1) node [above] $3$ – (24.5,1);
[thick,red] (24.5,0) – (25.5,2);
[thick] (25.5,1) –(26.5,1) node [above] $2$ – (27.5,1);
[thick,blue](28.5,0) – (27.5,2);
[thick](28.5,1) –(29.5,1) node [above] $2$ – (30.5,1);
[thick,blue](31.5,0) – (30.5,2);

[dashed, black](4.5,-.25) to [out=-45,in=225] (12.5,-.25);
[dashed, black](10.5,-.25) to [out=-45,in=225] (12.5,-.25);
[dashed, black](10.5,-.25) to [out=-45,in=225] (15.5,-.25);
[dashed, black](10.5,-.25) to [out=-45,in=225] (24.5,-.25);
[dashed, black](22.5,-.25) to [out=-45,in=225] (24.5,-.25);

[dashed, black](1.5,2.25) to [out=45,in=-225] (3.5,2.25);
[dashed, black](1.5,2.25) to [out=45,in=-225] (9.5,2.25);
[dashed, black](13.5,2.25) to [out=45,in=-225] (21.5,2.25);
[dashed, black](16.5,2.25) to [out=45,in=-225] (21.5,2.25);
[dashed, black](19.5,2.25) to [out=45,in=-225] (30.5,2.25);
[dashed, black](25.5,2.25) to [out=45,in=-225] (30.5,2.25);

An example of a tie diagram whose underlying brane diagram is the one displayed in Figure <ref>.

Next, we will create a correspondence between tie diagrams and binary contingency tables (BCTs). An intermediate notion will be required. An integer called the “charge" will be associated to each 5-brane.
For an NS5 brane $V$ let

\[
\charge(V)=
(d_{V^+}-d_{V^-})+\#\{\text{D5 branes left of $V$}\}.
\]

For a D5 brane $U$ let

\[
\charge(U)=
(d_{U^-}-d_{U^+})+\#\{\text{NS5 branes right of $U$}\}.
\]

Define the “row margin" and “column margin", respectively, by

\begin{align*}
r &= (\charge(V_1), ..., \charge(V_n)), \\
c &= (\charge(U_1), ..., \charge(U_m)).
\end{align*}

A “binary contingency table" (BCT) with row and column margins $r, c$ is a binary matrix with row sums equal to $r$ and columns sums equal to $c$. Note that the existence of such a matrix implies that $\sum_i r_i = \sum_j c_j$. Given a tie diagram, construct a BCT $M$ with margins $r, c$ as follows

* if $V_i$ is to the left of $U_j$, set $M_{ij} = 1$ if there is a tie between $V_i$ and $U_j$ and $M_{ij} = 0$ otherwise,

* if $V_i$ is to the right of $U_j$, set $M_{ij} = 1$ if there is no tie between $V_i$ and $U_j$ and $M_{ij} = 0$ otherwise.

Assuming the underlying brane diagram $\brane$ is fixed, this gives a bijection between tie diagrams and BCTs with margins $r, c$. However, the map sending all tie diagrams (with arbitrary underlying brane diagram) to their corresponding BCT is not injective. An additional piece of data, called the “separating line", is required to make this map bijective. A separating line is a path running from the top left to the bottom right of an $(m + 1) \times (n + 1)$ edit: corrected lattice dimensions lattice making only downward and rightward moves. We draw the separating line over the BCT, as depicted in Figure <ref>. The separating line plays the role of encoding the order of the 5-branes. Downward moves correspond to NS5 branes and rightward moves to D5 branes. A BCT along with a separating line is called a “table-with-margins". Table-with-margins are in bijection with tie diagrams.

[baseline=1.4 cm, scale=.6]
[ultra thin] (0,0) – (5,0);
[ultra thin]  (0,1) – (5,1);
[ultra thin]  (0,2) – (5,2);
[ultra thin]  (0,3) – (5,3);
[ultra thin]  (0,4) – (5,4);
[ultra thin]  (0,5) – (5,5);
[ultra thin]  (0,6) – (5,6);
[ultra thin]  (0,0) – (0,6);
[ultra thin]  (1,0) – (1,6);
[ultra thin]  (2,0) – (2,6);
[ultra thin]  (3,0) – (3,6);
[ultra thin]  (4,0) – (4,6);
[ultra thin]  (5,0) – (5,6);
[ultra thick] (0,6) – (0,5) – (1,5) – (1,4) – (2,4) – (2,1) – (3,1) – (3,0) – (5,0);
at (.5,6.4) $5$; at (1.5,6.4) $2$; at (2.5,6.4) $2$; at (3.5,6.4) $0$; at (4.5,6.4) $2$;
at (.5,7.4) $U_1$; at (1.5,7.4) $U_2$; at (2.5,7.4) $U_3$; at (3.5,7.4) $U_4$; at (4.5,7.4) $U_5$;
at (-.5,0.5) $2$; at (-.5,1.5) $3$; at (-.5,2.5) $2$; at (-.5,3.5) $1$; at (-.5,4.5) $1$; at (-.5,5.5) $2$;
at (-1.8,0.5) $V_6$; at (-1.8,1.5) $V_5$; at (-1.8,2.5) $V_4$; at (-1.8,3.5) $V_3$; at (-1.8,4.5) $V_2$; at (-1.8,5.5) $V_1$;
[violet] at (0.5,0.5) $1$;[violet] at (1.5,0.5) $0$;[violet] at (2.5,0.5) $0$;[violet] at (3.5,0.5) $0$;[violet] at (4.5,0.5) $1$;
[violet] at (0.5,1.5) $1$;[violet] at (1.5,1.5) $1$;[violet] at (2.5,1.5) $0$;[violet] at (3.5,1.5) $0$;[violet] at (4.5,1.5) $1$;
[violet] at (0.5,2.5) $1$;[violet] at (1.5,2.5) $0$;[violet] at (2.5,2.5) $1$;[violet] at (3.5,2.5) $0$;[violet] at (4.5,2.5) $0$;
[violet] at (0.5,3.5) $0$;[violet] at (1.5,3.5) $0$;[violet] at (2.5,3.5) $1$;[violet] at (3.5,3.5) $0$;[violet] at (4.5,3.5) $0$;
[violet] at (0.5,4.5) $1$;[violet] at (1.5,4.5) $0$;[violet] at (2.5,4.5) $0$;[violet] at (3.5,4.5) $0$;[violet] at (4.5,4.5) $0$;
[violet] at (0.5,5.5) $1$;[violet] at (1.5,5.5) $1$;[violet] at (2.5,5.5) $0$;[violet] at (3.5,5.5) $0$;[violet] at (4.5,5.5) $0$;

The table-with-margins corresponding to the tie diagram in Figure <ref>.

provides a more general construction of bow varieties of affine type A. In this more general setting, BCTs are replaced by the “Maya diagrams" of [6]. The precise relationship between BCTs and Maya diagrams is discussed in an appendix of [8].

Finally, we will describe butterfly diagrams. These provide a depiction of a representative in the prequotient $\prequot$ of a $\torus$-fixed point of $\bowvar(\brane)$. Fix a tie diagram and a D5 brane $U$. For each segment $X$, let $d_X^U$ be the number of distinct ties connected to $U$ covering $X$. Place a column of $d_X^U$ vertices below $X$ with fixed spacing, and align them in the following way:

* If the 5-brane between two consecutive columns is D5, align the columns at the bottom.

* For columns to the right of $U$, if the 5-brane between two consecutive columns is NS5, align the columns at the top.

* For columns to the left of $U$, if the 5-brane between two consecutive columns is NS5, align the columns so that the top vertex of the left column is one position lower than that of the right.

Also, place a special “framing" vertex (indicated by an open circle in our diagrams) below $U$. We will then join these vertices with directed edges:

* If a column is adjacent to a D5 brane, create downward (black) edges between each consecutive pair of vertices in the column.

* If the 5-brane between two adjacent columns is D5, create leftward (blue) edges between horizontally adjacent pairs of vertices.

* If the 5-brane between two adjacent columns is NS5, create (magenta dotted) edges pointing left one position and down one position, wherever possible, and rightward (red) edges between horizontally adjacent pairs of vertices.

* If $d_{U^-}^U > 0$, create an (green) edge from the framing vertex to the top vertex under $U^-$.

* If $d_{U^+}^U > d_{U^-}^U$, create an (green) edge from vertex $d_{U^+}^U - d_{U^-}^U$ (counted from top to bottom) to the framing vertex.

The resulting directed graph is called a “butterfly". Taking the disjoint union of the butterflies of all the D5 branes gives us a “butterfly diagram". Figure <ref> displays the butterfly diagram corresponding to the tie diagram in Figure <ref>.

[thick, red] (0.5,0) –(1.5,2);
[thick] (1,1)–(2.5,1) node [above] $2$ – (31,1);
[thick,blue](4.5,0) –(3.5,2);
[thick](4.5,1)–(5.5,1) node [above] $2$ – (6.5,1);
[thick,red](6.5,0) – (7.5,2);
[thick](7.5,1) –(8.5,1) node [above] $2$ – (9.5,1);
[thick,blue] (10.5,0) – (9.5,2);
[thick] (10.5,1) –(11.5,1) node [above] $4$ – (12.5,1);
[thick,red](12.5,0) – (13.5,2);
[thick](13.5,1) –(14.5,1) node [above] $3$ – (15.5,1);
[thick,red] (15.5,0) – (16.5,2);
[thick](16.5,1) –(17.5,1) node [above] $3$ – (18.5,1);
[thick,red](18.5,0) – (19.5,2);
[thick](19.5,1) –(20.5,1) node [above] $4$ – (21.5,1);
[thick,blue](22.5,0) – (21.5,2);
[thick](22.5,1) –(23.5,1) node [above] $3$ – (24.5,1);
[thick,red] (24.5,0) – (25.5,2);
[thick] (25.5,1) –(26.5,1) node [above] $2$ – (27.5,1);
[thick,blue](28.5,0) – (27.5,2);
[thick](28.5,1) –(29.5,1) node [above] $2$ – (30.5,1);
[thick,blue](31.5,0) – (30.5,2);

[dashed, black](4.5,-.25) to [out=-45,in=225] (12.5,-.25);
[dashed, black](10.5,-.25) to [out=-45,in=225] (12.5,-.25);
[dashed, black](10.5,-.25) to [out=-45,in=225] (15.5,-.25);
[dashed, black](10.5,-.25) to [out=-45,in=225] (24.5,-.25);
[dashed, black](22.5,-.25) to [out=-45,in=225] (24.5,-.25);

[dashed, black](1.5,2.25) to [out=45,in=-225] (3.5,2.25);
[dashed, black](1.5,2.25) to [out=45,in=-225] (9.5,2.25);
[dashed, black](13.5,2.25) to [out=45,in=-225] (21.5,2.25);
[dashed, black](16.5,2.25) to [out=45,in=-225] (21.5,2.25);
[dashed, black](19.5,2.25) to [out=45,in=-225] (30.5,2.25);
[dashed, black](25.5,2.25) to [out=45,in=-225] (30.5,2.25);

[fill] (2.5,-4) circle [radius=.04]; [fill] (5.5,-4) circle [radius=.04]; [fill] (8.5,-4) circle [radius=.04]; [fill] (11.5,-4) circle [radius=.04];
[blue, <-] (2.7,-4)–(5.3,-4); [red, ->] (5.7,-4)–(8.3,-4); [blue, <-] (8.7,-4)–(11.3,-4);
[<-,green] (2.5,-3.8) to [out=70,in=-120] (4,-3.1); [] (4,-3) circle [radius=.1];

[fill] (11.5,-5) circle [radius=.04]; [fill] (14.5,-5) circle [radius=.04]; [fill] (17.5,-5) circle [radius=.04]; [fill] (20.5,-5) circle [radius=.04]; [fill] (23.5,-5) circle [radius=.04];
[red, ->] (11.7,-5)–(14.3,-5);[red, ->] (14.7,-5)–(17.3,-5);[red, ->] (17.7,-5)–(20.3,-5);[blue,<-] (20.7,-5)–(23.3,-5);
[fill] (11.5,-6) circle [radius=.04]; [fill] (14.5,-6) circle [radius=.04];
[red, ->] (11.7,-6)–(14.3,-6);
[fill] (11.5,-7) circle [radius=.04];
[magenta, dotted,->] (14.25,-5.1) – (11.8,-5.9);[magenta, dotted,->] (17.25,-5.1) – (14.8,-5.9);[magenta, dotted,->] (14.25,-6.1) – (11.8,-6.9);
[blue,<-] (8.7,-7)–(11.3,-7);
[fill] (8.5,-7) circle [radius=.04];[fill] (5.5,-8) circle [radius=.04];[fill] (2.5,-8) circle [radius=.04];
[magenta, dotted,->] (8.25,-7.1) – (5.8,-7.9);[blue, ->] (5.25,-8) – (2.8,-8);
[->] (11.5,-6.1) – (11.5,-6.9);[->] (11.5,-5.1) – (11.5,-5.9);
[<-,green] (8.5,-6.8) to [out=70,in=-120] (9.9,-5.5); [<-,green] (10.1,-5.5) to [out=-50,in=160] (11.5,-6); [] (10,-5.5) circle [radius=.1];

[fill] (17.5,-8) circle [radius=.04]; [fill] (20.5,-8) circle [radius=.04]; [fill] (23.5,-8) circle [radius=.04];[fill] (20.5,-7) circle [radius=.04];[fill] (17.5,-9) circle [radius=.04];[fill] (14.5,-9) circle [radius=.04];
[red, ->] (14.7,-9)–(17.3,-9);[red, ->] (17.7,-8)–(20.3,-8);[blue, ->] (23.25,-8) – (20.8,-8);
[magenta, dotted,->] (20.25,-7.1) – (17.8,-7.9);[magenta, dotted,->] (20.25,-8.1) – (17.8,-8.9);[magenta, dotted,->] (17.25,-8.1) – (14.8,-8.9);
[->] (20.5,-7.1) – (20.5,-7.9);
[<-,green] (20.5,-6.8) to [out=70,in=-120] (21.9,-5.9);[] (21.9,-5.8) circle [radius=.1];

[fill] (20.5,-10) circle [radius=.04];[fill] (23.5,-10) circle [radius=.04];[fill] (26.5,-10) circle [radius=.04];[fill] (29.5,-10) circle [radius=.04];[fill] (26.5,-9) circle [radius=.04];[fill] (29.5,-9) circle [radius=.04];
[->] (29.5,-9.1) – (29.5,-9.9);[->] (26.5,-9.1) – (26.5,-9.9);
[blue, ->] (29.25,-10) – (26.8,-10);[blue, ->] (29.25,-9) – (26.8,-9);[blue, ->] (23.25,-10) – (20.8,-10);
[magenta, dotted,->] (26.25,-9.1) – (23.8,-9.9);
[red, ->] (23.7,-10)–(26.3,-10);
[<-,green] (29.5,-8.8) to [out=70,in=-120] (30.9,-7.9);[] (30.9,-7.8) circle [radius=.1];

The tie diagram from Figure <ref> along with its corresponding butterfly diagram.

For each segment $X$, pick a basis for $W_X$ and identify the basis vectors with the vertices of the butterfly diagram below $X$. For each D5 brane $U$, identify $\CC_U$ with the framing vertex of the butterfly of $U$. Then, interpret the edges as $A, -B, C, D, a, -b$ maps. Note that when $X^-$ and $X^+$ are both D5 branes, there are two $B$ maps acting on $W_X$, one from the triangle part of $X^-$ and one from the triangle part of $X^+$. Considering the $0$-momentum condition, we take these maps to be equal, both represented by the black edges below $X$. The maps determined by the butterfly diagram constitute an element of $\prequot$. Moreover, this element descends to a $\torus$-fixed point of $\bowvar(\brane)$. The converse is also true, that is, every $\torus$-fixed point of $\bowvar(\brane)$ can be represented by an element of $\prequot$ determined by some butterfly diagram. This is the content of <cit.>.

§.§ Fixed Point Restrictions

Recall that a bow variety $\bowvar(\brane)$ comes equipped with tautological bundles $\xi_X$, one for each segment. Let $p\in \bowvar(\brane)^\torus$, and consider the corresponding butterfly diagram. Each non-framing vertex will be assigned a “height" relative to the butterfly it belongs to.

Fix a butterfly and define the “height" of each of its vertices as follows:

* If the tail of a green edge is the framing vertex, the head is a vertex of height 0.

* If the head of a green edge is the framing vertex, the tail is a vertex of height 1.

* The height decreases by 1 when we go from one vertex to another one immediately below it, and is constant across horizontal rows of vertices.

Given any vertex $v$ in a butterfly diagram, denote its height by $y(v)$. Note that the height of $v$ is measured with respect to the butterfly containing $v$.

Let $\ub$ be the trivial line bundle over $p$ whose $\torus$-action is given by scaling by the $\CC_U$ factor. Similarly, let $\hb$ be the trivial line bundle scaled by $\CC_\hbar$. Decorate each vertex $v$ of the $U$ butterfly by $\ub \hb^{y(v)}\in K^0_\torus(p)$. Figure <ref> shows these decorations for the butterfly diagram of Figure <ref>. Then, we have that $\xi_X|_p$ is equal the sum of the line bundles decorating the vertices of the butterfly diagram below $X$ as elements of $K^0_\torus(p)$. For example, in Figure <ref>, we have

\[
\xi_{X_7} = \ub_2\hb^2 + \ub_3 + \frac{\ub_3}{\hb} + \frac{\ub_4}{\hb}.
\]

[scale=.5, yscale=2]
(2.5,-4) node(1a1)$\ub_1$; [fill] (5.5,-4) node(2a1)$\ub_1$ (2a1); [fill] (8.5,-4) node(3a1)$\ub_1$;
[fill] (11.5,-4) node(4a1)$\ub_1$;
[blue, <-] (1a1)–(2a1); [red, ->] (2a1)–(3a1); [blue, <-] (3a1)–(4a1);
(4, -3) node [draw, circle, inner sep = 0, minimum size = 0.1cm] (u1) ; [<-,green] (1a1) to [out=70,in=-120] (u1);

[fill] (11.5,-5) node(4b1)$\ub_2\hb^{2}$; [fill] (14.5,-5) node(5b1)$\ub_2\hb^{2}$;
[fill] (17.5,-5) node(6b1)$\ub_2\hb^2$; [fill] (20.5,-5) node(7b1)$\ub_2\hb^2$;
[fill] (23.5,-5) node(8b1)$\ub_2\hb^2$;
[red, ->] (4b1)–(5b1);[red, ->] (5b1)–(6b1);[red, ->] (6b1)–(7b1);[blue,<-] (7b1)–(8b1);
[fill] (11.5,-6) node(4b2)$\ub_2\hb$; [fill] (14.5,-6) node(5b2)$\ub_2\hb$;
[red, ->] (4b2)–(5b2);
[fill] (11.5,-7) node(4b3)$\ub_2$;
[magenta, dotted,->] (5b1) – (4b2);[magenta, dotted,->] (6b1) – (5b2);[magenta, dotted,->] (5b2) – (4b3);
[fill] (8.5,-7) node(3b1)$\ub_2$;[fill] (5.5,-8) node(2b1)$\frac{\ub_2}{\hb}$;
[fill] (2.5,-8) node(1b1)$\frac{\ub_2}{\hb}$;
[blue,<-] (3b1)–(4b3);[magenta, dotted,->] (3b1) – (2b1);[blue, ->] (2b1) – (1b1);
[->] (4b1) – (4b2);[->] (4b2) – (4b3);
[] (10,-5.5) node [draw, circle, inner sep = 0, minimum size = 0.1cm] (u2) ;
[<-,green] (3b1) to [out=70,in=-120] (u2); [<-,green] (u2) to [out=-50,in=160] (4b2);

[fill] (17.5,-8) node(2c1)$\frac{\ub_3}{\hb}$; [fill] (20.5,-8) node(3c2)$\frac{\ub_3}{\hb}$;
[fill] (23.5,-8) node(4c1)$\frac{\ub_3}{\hb}$;
[fill] (20.5,-7) node(3c1)$\ub_3$;
[fill] (17.5,-9) node(2c2)$\frac{\ub_2}{\hb^2}$;[fill] (14.5,-9) node(1c1)$\frac{\ub_2}{\hb^2}$;
[red, ->] (1c1)–(2c2);[red, ->] (2c1)–(3c2);[blue, ->] (3c2) – (4c1);
[magenta, dotted,->] (2c1) – (1c1);[magenta, dotted,->] (3c1) – (2c1);[magenta, dotted,->] (3c2) – (2c2);
[->] (3c1) – (3c2);
[] (21.9,-5.8) node [draw, circle, inner sep = 0, minimum size = 0.1cm] (u3) ;[<-,green] (3c1) to [out=70,in=-120] (u3);

[fill] (20.5,-10) node(1d1)$\frac{\ub_4}{\hb}$;[fill] (23.5,-10) node(2d1)$\frac{\ub_4}{\hb}$;
[fill] (26.5,-10) node(3d2)$\frac{\ub_4}{\hb}$;[fill] (29.5,-10) node(4d2)$\frac{\ub_4}{\hb}$;
[fill] (26.5,-9) node(3d1)$\ub_4$;[fill] (29.5,-9) node(4d1)$\ub_4$;
[->] (3d1) – (3d2);[->] (4d1) – (4d2);
[blue, ->] (4d1) – (3d1);[blue, ->] (4d2) – (3d2);[blue, ->] (2d1) – (1d1);
[magenta, dotted,->] (3d1) – (2d1);
[red, ->] (2d1)–(3d2);
[] (30.9,-7.8) node [draw, circle, inner sep = 0, minimum size = 0.1cm] (u4) ;[<-,green] (4d1) to [out=70,in=-120] (u4);

The butterfly diagram of Figure <ref> decorated by elements of $K^0_\torus(p)$.

Of central importance in this work is the $\torus$-equivariant K-theory class of $T_p\bowvar(\brane)$. To express this class in terms of $\ub$ and $\hb$ bundles, one simply takes the tangent bundle formula (<ref>) and substitutes each tautological bundle with the formula for its fixed point restriction calculated as above. Then, one can apply bilinearity of $\Hom$ and the isomorphism $\Hom(\alpha, \beta)\cong \beta\otimes\alpha^*$ to expand this expression into a sum of line bundles. In our running example, after a great deal of cancellations, we obtain

\begin{multline}
\label{eqn:cancel}
T_p\bowvar(\brane) = \frac{\ub_2\hb^4}{\ub_5} + \frac{\ub_2\hb^4}{\ub_3} + \frac{\ub_1\hb^3}{\ub_3} + \frac{\ub_2\hb^3}{\ub_3}
+ \frac{\ub_4\hb^2}{\ub_5} + \frac{\ub_3\hb}{\ub_5} + \frac{\ub_4\hb}{\ub_5} + \frac{\ub_5\hb}{\ub_3} \\
+ \frac{\ub_3}{\ub_5}
+ \frac{\ub_5}{\ub_4} + \frac{\ub_5}{\ub_4\hb} + \frac{\ub_3}{\ub_2\hb^2} + \frac{\ub_3}{\ub_1\hb^2} + \frac{\ub_3}{\ub_2\hb^3}
+ \frac{\ub_5}{\ub_2\hb^3}.
\end{multline}

Each line bundle in this expansion is called a “tangent weight at $p$". In other words, the tangent weights at $p$ are the the K-theoretic Chern roots of $T_p\bowvar(\brane)$.

§.§ Hanany-Witten Transition and Separated Bow Varieties

Let $\brane$ contain a D5 brane $U$ followed immediately by an NS5 brane $V$. Let $\widetilde\brane$ agree with $\brane$ except at $U$ and $V$, where they differ as in the picture

\[
\begin{tikzpicture}[baseline=(current  bounding  box.center), scale=.5]
\draw[thick] (0,1)--(1,1) node [above] {$d_1$} -- (2,1);
\draw[thick,blue] (3,0)--(2,2);
\draw[thick] (3,1)--(4,1) node [above] {$d_2$}--(5,1);
\draw[thick,red] (5,0)--(6,2);
\draw[thick] (6,1)--(7,1) node [above] {$d_3$}--(8,1);
\node at (2.7,-.45) {$U$}; \node at (5.3,-.45) {$V$};
\node at (4,.3) {$X$};
%\draw[ultra thick, <->] (10,1)--(11.5,1) node[above]{HW} -- (13,1);
\draw[thick] (15,1)--(16,1) node [above] {$d_1$} -- (17,1);
\draw[thick,red] (17,0)--(18,2);
\draw[thick] (18,1)--(19,1) node [above] {$\widetilde{d}_2$}--(20,1);
\draw[thick,blue] (21,0)--(20,2);
\draw[thick] (21,1)--(22,1) node [above] {$d_3$}--(23,1);
\node at (20.7,-.4) {$\widetilde{U}$}; \node at (17.3,-.45) {$\widetilde{V}$};
\node at (19,.3) {$\widetilde{X}$};
\end{tikzpicture},
\]

where $d_2 + \widetilde d_2 = d_1 + d_3 + 1$. Then, there is a natural holomorphic symplectic isomorphism $\bowvar(\brane) \overset{\cong}{\to} \bowvar(\widetilde\brane)$ called a “Hanany-Witten transition". In other words, Hanany-Witten transitions allow one to exchange two consecutive 5-branes of different type (c.f. <cit.>) at the expense of changing the multiplicity of the D3 brane in between. The Hanany-Witten transition is equivariant with respect to the reparametrization $\widetilde \ub = \ub\hb$ (the other factors of the torus are not affected).

The Hanany-Witten transition induces a bijection of $\torus$-fixed points. This bijection can best be understood through transformations of tie diagrams and tables-with-margins. The action of the Hanany-Witten transition on tie diagrams is described by Figure <ref>. In words, switch $U$ and $V$ while keeping all ties attached. If there was a tie between $U$ and $V$, remove it, and if there was no such tie, create one.

[thick] (0,1) –(2,1);
[thick, blue] (3,0) – (2,2);
[thick] (3,1)–(5,1);
[thick,red] (5,0)–(6,2);
[thick] (6,1)–(8,1);
[dashed](2,2) to [out=120,in=0] (0,3) ;
[dashed](2,2) to [out=120,in=0] (0,3.25) node [left] $A$ ;
[dashed](2,2) to [out=120,in=0] (0,3.4) ;
[dashed](5,0) to [out=240,in=0] (0,-2)  ;
[dashed](5,0) to [out=240,in=0] (0,-2.25)  node [left] $B$ ;
[dashed](5,0) to [out=240,in=0] (0,-2.4);
[dashed](3,0) to [out=-60,in=180] (8,-2);
[dashed](3,0) to [out=-60,in=180] (8,-2.25) node [right]$C$;
[dashed](3,0) to [out=-60,in=180] (8,-2.4) ;
[dashed](6,2) to [out=60,in=180] (8,3) ;
[dashed](6,2) to [out=60,in=180] (8,3.25) node [right]$D$;
[dashed](6,2) to [out=60,in=180] (8,3.4);
[dashed] (3,0) to [out=-40,in=-140](5,0);
at (4,0.25) $E$;

[scale=.5, baseline=-30pt]
[ultra thick, <->] (0,1)–(2,1) node[above]HW transition – (4,1);
[ultra thick, <->] (0,1)–(2,1) node[below]on fixpoints – (4,1);

[thick] (0,1) –(2,1);
[thick, red] (2,0) – (3,2);
[thick] (3,1)–(5,1);
[thick, blue] (6,0)–(5,2);
[thick] (6,1)–(8,1);
[dashed](2,0) to [out=240,in=0] (0,-.7);
[dashed](2,0) to [out=240,in=0] (0,-.9)  node [left] $B$;
[dashed](2,0) to [out=240,in=0] (0,-1.1) ;
[dashed](5,2) to [out=120,in=0] (0,4)  ;
[dashed](5,2) to [out=120,in=0] (0,4.25)  node [left] $A$;
[dashed](5,2) to [out=120,in=0] (0,4.4) ;
[dashed](3,2) to [out=60,in=180] (8,4);
[dashed](3,2) to [out=60,in=180] (8,4.25) node [right]$D$;
[dashed](3,2) to [out=60,in=180] (8,4.4) ;
[dashed](6,0) to [out=-60,in=180] (8,-.7) ;
[dashed](6,0) to [out=-60,in=180] (8,-.9)node [right]$C$;
[dashed](6,0) to [out=-60,in=180] (8,-1.1);
[dashed] (3,2) to [out=40,in=140](5,2);
at (4,1.8) $\lnot E$;

The action of Hanany-Witten transition on tie diagrams. The $E$ and $\neg E$ symbols mean that the tie labeled $E$ is present if and only if the tie labeled $\neg E$ is not.

The Hanany-Witten transition acts on tables-with-margins by perturbing the separating line while leaving the BCT fixed.

By a sequence of Hanany-Witten transitions, one may transform any brane diagram into a brane diagram with all NS5 branes on the left and all D5 branes on the right. Such a brane diagram will be called “separated", as will the associated bow variety. Figure <ref> depicts the result of applying this procedure to separate the tie diagram of Figure <ref>.

[scale = 0.5]

[thick,red] (-0.5,-1)–(0.5,1) node[pos=0.5](V1) node[pos=1,inner sep=0](V1t) node[pos=0,inner sep=0](V1b);
[thick,red] (2.5,-1)–(3.5,1) node[pos=0.5](V2) node[pos=1,inner sep=0](V2t) node[pos=0,inner sep=0](V2b);
[thick,red] (5.5,-1)–(6.5,1) node[pos=0.5](V3) node[pos=1,inner sep=0](V3t) node[pos=0,inner sep=0](V3b);
[thick,red] (8.5,-1)–(9.5,1) node[pos=0.5](V4) node[pos=1,inner sep=0](V4t) node[pos=0,inner sep=0](V4b);
[thick,red] (11.5,-1)–(12.5,1) node[pos=0.5](V5) node[pos=1,inner sep=0](V5t) node[pos=0,inner sep=0](V5b);
[thick,red] (14.5,-1)–(15.5,1) node[pos=0.5](V6) node[pos=1,inner sep=0](V6t) node[pos=0,inner sep=0](V6b);
[thick,blue] (18.5,-1)–(17.5,1) node[pos=0.5](U1) node[pos=1,inner sep=0](U1t) node[pos=0,inner sep=0](U1b);
[thick,blue] (21.5,-1)–(20.5,1) node[pos=0.5](U2) node[pos=1,inner sep=0](U2t) node[pos=0,inner sep=0](U2b);
[thick,blue] (24.5,-1)–(23.5,1) node[pos=0.5](U3) node[pos=1,inner sep=0](U3t) node[pos=0,inner sep=0](U3b);
[thick,blue] (27.5,-1)–(26.5,1) node[pos=0.5](U4) node[pos=1,inner sep=0](U4t) node[pos=0,inner sep=0](U4b);
[thick,blue] (30.5,-1)–(29.5,1) node[pos=0.5](U5) node[pos=1,inner sep=0](U5t) node[pos=0,inner sep=0](U5b);

(V1)–(V2) node[pos=0.5, above](X1)2;
(V2)–(V3) node[pos=0.5, above](X2)3;
(V3)–(V4) node[pos=0.5, above](X3)4;
(V4)–(V5) node[pos=0.5, above](X4)6;
(V5)–(V6) node[pos=0.5, above](X5)9;
(V6)–(U1) node[pos=0.5, above](X6)11;
(U1)–(U2) node[pos=0.5, above](X7)6;
(U2)–(U3) node[pos=0.5, above](X8)4;
(U3)–(U4) node[pos=0.5, above](X9)2;
(U4)–(U5) node[pos=0.5, above](X10)2;

[dashed] (V1t) to [out=55,in=125] (U1t);
[dashed] (V1t) to [out=57,in=123] (U2t);
[dashed] (V3t) to [out=55,in=125] (U3t);
[dashed] (V4t) to [out=53,in=127] (U3t);
[dashed] (V5t) to [out=55,in=125] (U5t);
[dashed] (V6t) to [out=53,in=127] (U5t);
[dashed] (V2t) to [out=53,in=127] (U1t);
[dashed] (V4t) to [out=49,in=131] (U1t);
[dashed] (V5t) to [out=49,in=131] (U2t);
[dashed] (V5t) to [out=47,in=133] (U1t);
[dashed] (V6t) to [out=45,in=135] (U1t);

The tie diagram of Figure <ref> put into separated form using Hanany-Witten transitions.

Separated bow varieties will be the focus of the remainder of this work. Any results derived can be translated to non-separated bow varieties by making a suitable reparametrization of $\torus$. In practical terms, this amounts to adjusting the exponents of $\hb$ in the formulas.

§ A COMBINATORIAL FORMULA FOR TANGENT WEIGHTS

Fix a separated brane diagram $\brane$ and torus fixed point $p\in\bowvar(\brane)^\torus$. We will give a combinatorial formula for the tangent weights at $p$ (i.e. K-theoretic Chern roots of $T_p\bowvar(\brane)$) in terms of the associated table-with-margins. Unlike the approach used to derive (<ref>), this formula will return the tangent weights without the need for any further cancellations or simplifications. Note that since the bow variety is separated, the separating line always runs along the left and bottom sides. We ignore it and consider only the BCT $M$. Again, since the bow variety is separated, we have

\[
M_{ij} = \begin{cases}
1 & \text{if there is a tie between }V_i\text{ and }U_j, \\
0 & \text{otherwise}. \end{cases}
\]

For $0\leq i \leq n, 1 \leq j \leq m$, define

\[
s_{ij} = \sum_{i' = 1}^i M_{i' j}.
\]

Furthermore, define a “$kl$-pair" to be a tuple of indices $(i, j, j')$ such that $M_{i j} = k$, $M_{i j'} = l$, and $j < j'$.

Given a separated brane diagram $\brane$ and a fixed point $p\in\bowvar(\brane)^\torus$, we have

\begin{equation}
\label{eqn:bct_weights}
T_p\bowvar(\brane) = \sum_{(i, j_0, j_1)\text{ 01-pair}} \left(
\frac{\ub_{j_0}}{\ub_{j_1}}\hb^{s_{i j_1}-s_{i j_0}} + \frac{\ub_{j_1}}{\ub_{j_0}}\hb^{s_{i j_0}-s_{i j_1}+1} \right)
\end{equation}

as elements of $K^0_\torus(p)$.

To aid in our proof, let us establish some notation. For $1\leq j \leq m$ and $r\in\NN$, let

\[
r*\ub_j = \ub_j + \frac{\ub_j}{\hb} + \frac{\ub_j}{\hb^2} + \cdots + \frac{\ub_j}{\hb^{r-1}} \in K^0_\torus(p).
\]

Using this notation, we can rephrase the formula for the fixed point restriction of tautological bundles of Section <ref> as

\begin{equation}
\label{eqn:bct_res}
\xi_k|_p = \begin{cases}
\sum_j \hb^{k-n}(s_{kj}*\ub_j) & \text{if }k\leq n, \\
\sum_j s_{nj}*\ub_j & \text{if }n\leq k \leq n+m-1, \\
0 & \text{otherwise}.
\end{cases}
\end{equation}

It will also be helpful to summarize the tangent bundle formula (<ref>) using the diagram

\begin{equation}
\label{eqn:tangent_sep}
\small
\begin{tikzcd}[column sep = 1.1em]
\xi_1 \ar[rr, bend left, "1"] \cdloop{-1-\hb}
&& \cdots \ar[ll, bend left, "\hb"] \ar[rr, bend left, "1"]
&& \xi_{n - 1} \ar[ll, bend left, "\hb"] \ar[rr, bend left, "1"] \cdloop{-1-\hb}
&& \xi_n \ar[ll, bend left, "\hb"] \cdloop{-1}
&& \xi_{n + 1} \ar[ll, swap, "1-\hb"] \cdloop{-1+\hb} \ar[ld, "\hb"]
&& \cdots \ar[ll, swap, "1-\hb"] \ar[ld, "\hb"]
&& \xi_{n+m-1} \ar[ll, swap, "1-\hb"] \cdloop{-1+\hb} \ar[ld, "\hb"]
\\
\ub_{1} \ar[lu, "1"]
&& \ub_{2} \ar[lu, "1"]
&& \ub_{{m-1}} \ar[lu, "1"]
&& \ub_{m} \ar[lu, "1"]
\end{tikzcd},
\end{equation}

where $\xi_k = \xi_{X_k}$.

We will induct on $\sum_{i, j} M_{ij}$, the number of 1's in $M$. It will be convenient to divide formula (<ref>) into four types of parts. These parts can also be respresented by pieces of diagram (<ref>).

ξ_i-1 -1- [r, bend left, "1"]
ξ_i [l, bend left, ""]

$P_{1i} = \Hom(\xi_{i-1}, \xi_{i}) + \hb\Hom(\xi_{i}, \xi_{i-1}) + (-1-\hb)\End(\xi_{i-1})$ for $1\leq i \leq n$.

ξ_n+j [l, "1-"] -1+

$P_{2j} = (1-\hb)\Hom(\xi_{n+j}, \xi_{n+j-1}) + (-1+\hb)\End(\xi_{n+j})$ for $1 \leq j\leq m$.

ξ_n+j-1        ξ_n+j [ld, ""]

_j [lu, "1"]

$P_{3j} = \Hom(\ub_j, \xi_{n+j-1}) + \hb\Hom(\xi_{n+j}, \ub_j)$ for $1\leq j\leq m$, where $\xi_{n+m}=0$ by convention.

ξ_n -1

$P_4 = -\End(\xi_n)$.

We begin with the base case, where there is a single 1 in position $(i_1, j_1)$ and 0's elsewhere. This corresponds to a tie diagram with a single tie joining $V_{i_1}$ with $U_{j_1}$. From the height formula of Section <ref>, we have

\begin{equation} \label{eqn:base_res}
\xi_k|_p = \begin{cases}
\ub_{j_1} \hb^{k - n} & \text{if }i_1\leq k\leq n, \\
\ub_{j_1} & \text{if }n\leq k\leq n + j_1 - 1, \\
0 & \text{otherwise.}
\end{cases}
\end{equation}

It is easily verified that all $P_{1i}$ and $P_{2j}$ parts vanish upon restriction to $p$. Consider now the terms of $P_{3j}|_p$. For $j < j_1$, we obtain $\frac{\ub_{j_1}}{\ub_j} + \frac{\ub_j}{\ub_{j_1}}\hb$. For $j > j_1$, we obtain 0. Finally, for $j = j_1$, we obtain $1$, which cancels with the remaining part $P_4|_p$. This proves the base case.

Now, we prove the inductive step. Given $M$, let $i_1$ be the maximum index for which row $i_1$ has a 1. Then, let $j_1$ be the maximum index for which $M_{i_1 j_1} = 1$. It follows that

\begin{equation}
\label{eqn:s_max}
s_{ij}=s_{i_1 j} \text{ for all } 1\leq j\leq m \text{ and } i\geq i_1.
\end{equation}

The inductive step will be proven by analyzing the effect of switching $M_{i_1 j_1}$ from 1 to 0, or equivalently, deleting the tie joining $V_{i_1}$ with $U_{j_1}$. By assumption, this tie is the shortest tie among those connected to $U_{j_1}$. Therefore, the effect of deleting it is to shave off the bottom layer of vertices between $X_{i_1}$ and $X_{n+j_1-1}$ from the $U_{j_1}$ butterfly. The other butterflies are unaffected. Let $p' \in\bowvar(\brane')$ be the fixed point associated with the tie diagram obtained by deleting this tie. Define the set

\[
\kappa=\{\ub_1^{\pm 1}, ..., \ub_m^{\pm 1}, \xi_0, ..., \xi_{n+m}\}\subset K^0_\torus(\bowvar(\brane)).
\]

For any $\eta\in\kappa$, let $\Delta\eta = \eta|_p - \eta|_{p'} \in K^0_\torus(\pt)$. Similarly, for $\eta_1,\eta_2\in\kappa$, let $\Delta\Hom(\eta_1, \eta_2) = \Hom(\eta_1, \eta_2)|_p-\Hom(\eta_1, \eta_2)|_{p'}$. Note that

\begin{align*}
\Delta\Hom(\eta_1, \eta_2) &= \Hom(\Delta\eta_1, \eta_2|_p - \Delta\eta_2) + \Hom(\eta_1|_p - \Delta\eta_1, \Delta\eta_2) \\
&= \sum_{j\neq j_1} \left( \Hom \left( \Delta\eta_1, \eta_2^{(j)} \right) + \Hom \left( \eta_1^{(j)}, \Delta\eta_2 \right) \right) \\
&\hspace{70pt} +\Hom \left( \Delta\eta_1, \eta_2^{(j_1)} \right) + \Hom \left( \eta_1^{(j_1)}, \Delta\eta_2 \right)-2\Hom(\Delta\eta_1, \Delta\eta_2),
\end{align*}

where $\eta^{(j)}$ is the sum of the Chern roots of $\eta|_p$ of the form $\ub_j \hb^r$. For $1\leq j\leq m$, let

\[
\Delta^{(j)}\Hom(\xi_{k_1}, \xi_{k_2}) = \begin{cases}
\Hom \left( \Delta\xi_{k_1}, \xi_{k_2}^{(j)} \right) + \Hom \left( \xi_{k_1}^{(j)}, \Delta\xi_{k_2} \right) & \text{if }j\neq j_1, \\
\Hom \left( \Delta\xi_{k_1}, \xi_{k_2}^{(j_1)} \right) + \Hom \left( \xi_{k_1}^{(j_1)}, \Delta\xi_{k_2} \right)-2\Hom(\Delta\xi_{k_1}, \Delta\xi_{k_2})
&\text{if }j=j_1.
\end{cases}
\]

Extend the definitions of $\Delta$ and $\Delta^{(j)}$ to the vector subspace $\CC\cdot\kappa\subset K^0_\torus(\bowvar(\brane))$ by linearity. Note that $\Delta = \sum_j \Delta^{(j)}$ as linear maps.

From (<ref>) and (<ref>), we see that

\[
\Delta\xi_k = \begin{cases}
\ub_{j_1}\hb^{k-n-s_{i_1j_1}+1} & \text{if }i_1\leq k\leq n, \\
\ub_{j_1}\hb^{-s_{i_1j_1}+1} & \text{if }n \leq k\leq n+j_1 -1, \\
0 & \text{otherwise}
\end{cases}
\]

(c.f. (<ref>)). Fix $j_0\neq j_1$. Because $\Delta\xi_k=0$ for $i<i_1$, we have $\Delta^{(j_0)}P_{1i}=0$ for $i<i_1$. By assumption, no ties end at $V_i$ for $i>i_1$. Hence, $\xi_k=\hb^{-1}\xi_{k+1}$ for $i_1\leq k \leq n-1$. It follows easily that $\Delta^{(j_0)}P_{1i}=0$ for $i_1< i$. Hence, the only $\Delta^{(j_0)}P_{1i}$ that can be nonzero is $\Delta^{(j_0)}P_{1 i_1}$. We have

\begin{equation}
\label{eqn:P1}
\begin{aligned}
\Delta^{(j_0)}P_{1 i_1} &= \Hom(\xi_{i_1 -1}^{(j_0)}, \Delta\xi_{i_1})+\hb\Hom(\Delta\xi_{i_1}, \xi_{i_1 -1}^{(j_0)}) \\
&= \Hom(\hb^{i_1-n-1}(s_{i_1-1, j_0}*\ub_{j_0}),\ub_{j_1}\hb^{i_1-n-s_{i_1 j_1}+1})\\
&\hspace{50pt}+\hb\Hom(\ub_{j_1}\hb^{i_1-n-s_{i_1 j_1}+1}, \hb^{i_1-n-1}(s_{i_1-1, j}*\ub_{j_0})) \\
&= \hb^{2-s_{i_1 j_1}}\Hom(s_{i_1-1, j_0}*\ub_{j_0},\ub_{j_1})+\hb^{s_{i_1 j_1}-1}\Hom(\ub_{j_1},s_{i_1-1, j_0}*\ub_{j_0}).
\end{aligned}
\end{equation}

Next, we consider $\Delta^{(j_0)}P_{2j}$ for $j_0\neq j_1$. For $j<\min\{j_0,j_1\}$, we have $\xi_{n+j-1}^{(j_0)}=\xi_{n+j}^{(j_0)}$ and $\Delta\xi_{n+j-1}=\Delta\xi_{n+j}$. It is easy to see that in this case $\Delta^{(j_0)}P_{2j}=0$. For $j>\min\{j_0,j_1\}$, we have $\Delta\xi_{n+j-1}=\Delta\xi_{n+j}=0$ or $\xi_{n+j-1}^{(j_0)}=\xi_{n+j}^{(j_0)}=0$, so $\Delta^{(j_0)}P_{2j}=0$. Finally, we have

\begin{equation}
\label{eqn:P2}
\begin{aligned}
\Delta^{(j_0)}P_{2\min\{j_0, j_1\}} &= \begin{cases}
(1-\hb)\Hom(\ub_{j_1}\hb^{-s_{i_1 j_1}+1}, s_{i_1 j_0}*\ub_{j_0}) & \text{if }j_0<j_1, \\
(1-\hb)\Hom(s_{i_1 j_0}*\ub_{j_0}, \ub_{j_1}\hb^{-s_{i_1 j_1}+1}) & \text{if }j_0>j_1.
\end{cases} \\
&= \begin{cases}
(\hb^{s_{i_1 j_1}-1}-\hb^{s_{i_1 j_1}})\Hom(\ub_{j_1}, s_{i_1 j_0}*\ub_{j_0}) & \text{if }j_0<j_1, \\
(\hb^{-s_{i_1 j_1}+1}-\hb^{2-s_{i_1 j_1}})\Hom(s_{i_1 j_0}*\ub_{j_0}, \ub_{j_1}) & \text{if }j_0>j_1.
\end{cases}
\end{aligned}
\end{equation}

Turning our attention to the $P_{3j}$ and $P_4$ parts, note that $\Delta^{(j_0)}P_{3j}$ vanishes trivially if $j\neq j_0$. We have

\begin{equation}
\label{eqn:P3}
\begin{aligned}
\Delta^{(j_0)}P_{3j_0} &= \begin{cases}
\Hom(\ub_{j_0}, \ub_{j_1}\hb^{-s_{i_1 j_1}+1}) + \hb\Hom(\ub_{j_1}\hb^{-s_{i_1 j_1}+1}, \ub_{j_0}) & \text{if }j_0<j_1, \\
0 & \text{if }j_0>j_1.
\end{cases} \\
&= \begin{cases}
\hb^{-s_{i_1 j_1}+1}\Hom(\ub_{j_0}, \ub_{j_1}) + \hb^{s_{i_1 j_1}}\Hom(\ub_{j_1}, \ub_{j_0}) & \text{if }j_0<j_1, \\
0 & \text{if }j_0>j_1.
\end{cases}
\end{aligned}
\end{equation}

\begin{equation}
\label{eqn:P4}
\begin{aligned}
\Delta^{(j_0)}P_4 &= -\Hom(\ub_{j_1}\hb^{-s_{i_1 j_1}+1}, s_{i_1 j_0}*\ub_{j_0}) -\Hom(s_{i_1 j_0}*\ub_{j_0}, \ub_{j_1}\hb^{-s_{i_1 j_1}+1}) \\
&= -\hb^{s_{i_1 j_1}-1}\Hom(\ub_{j_1}, s_{i_1 j_0}*\ub_{j_0}) -\hb^{-s_{i_1 j_1}+1}\Hom(s_{i_1 j_0}*\ub_{j_0}, \ub_{j_1}).
\end{aligned}
\end{equation}

Compute $\Delta^{(j_0)}T\bowvar(\brane)$ by adding together (<ref>), (<ref>), (<ref>), and (<ref>). If $j_0<j_1$, the result is

\begin{align*}
\Delta^{(j_0)}T\bowvar(\brane) &= \hb^{2-s_{i_1 j_1}}\Hom(s_{i_1-1, j_0}*\ub_{j_0}, \ub_{j_1}) - \hb^{-s_{i_1 j_1}+1}\Hom(s_{i_1 j_0}*\ub_{j_0}, \ub_{j_1}) \\
&\qquad +\hb^{s_{i_1 j_1}-1}\Hom(u_{j_1}, s_{i_1-1, j_0}*\ub_{j_0}) - \hb^{s_{i_1 j_1}}\Hom(\ub_{j_1}, s_{i_1, j_0}*\ub_{j_0}) \\
&\qquad +\hb^{-s_{i_1 j_1}+1}\Hom(\ub_{j_0}, \ub_{j_1}) + \hb^{s_{i_1 j_1}}\Hom(\ub_{j_1}, \ub_{j_0}) \\
&= \hb^{2-s_{i_1 j_1}}\Hom(s_{i_1-1, j_0}*\ub_{j_0}-\hb(s_{i_1 j_0}*\ub_{j_0}), \ub_{j_1}) \\
&\qquad +\hb^{s_{i_1 j_1}-1}\Hom(\ub_{j_1}, s_{i_1-1, j_0}*\ub_{j_0}-\hb(s_{i_1 j_0}*\ub_{j_0})) \\
&\qquad +\hb^{-s_{i_1 j_1}+1}\Hom(\ub_{j_0}, \ub_{j_1}) + \hb^{s_{i_1 j_1}}\Hom(\ub_{j_1}, \ub_{j_0}) \\
&= \hb^{2-s_{i_1 j_1}}\Hom((1-M_{i_1 j_0})\ub_j\hb^{-s_{i_1 j_0}+1}-\ub_{j_0}\hb, \ub_{j_1}) \\
&\qquad +\hb^{s_{i_1 j_1}-1}\Hom(\ub_{j_1}, (1-M_{i_1 j_0})\ub_j\hb^{-s_{i_1 j_0}+1}-\ub_{j_0}\hb) \\
&\qquad +\hb^{2-s_{i_1 j_1}}\Hom(\ub_{j_0}\hb, \ub_{j_1}) + \hb^{s_{i_1 j_1}-1}\Hom(\ub_{j_1}, \ub_{j_0}\hb) \\
&= (1-M_{i_1 j_0})\hb^{2-s_{i_1 j_1}}\Hom(\ub_{j_0}\hb^{-s_{i_1 j_0}+1}, \ub_{j_1}) \\
&\qquad +(1-M_{i_1 j_0})\hb^{s_{i_1 j_1}-1}\Hom(\ub_{j_1}, \ub_{j_0}\hb^{-s_{i_1 j_0}+1}) \\
&= (1-M_{i_1 j_0})\left( \frac{\ub_{j_1}}{\ub_{j_0}}\hb^{1+s_{i_1 j_0}-s_{i_1 j_1}}+\frac{\ub_{j_0}}{\ub_{j_1}}\hb^{s_{i_1 j_1}-s_{i_1 j_0}} \right),
\end{align*}

Where the third equality follows from the fact that $s_{i_1-1,j_0}=s_{i_1 j_0}-M_{i_1 j_0}$. If $j_0>j_1$, it is easy to see that all terms cancel. Hence, if $(i_1, j_0, j_1)$ is a 01-pair, then we get the two associated tangent weights that appear in the summation in (<ref>), and we get 0 otherwise.

Lastly, we must consider the case $j_0=j_1$. We find, again, that $\Delta^{(j_1)}P_{1i}$ vanishes unless $i=i_1$. In this case,

\begin{align*}
\Delta^{(j_1)}P_{1 i_1} &= \Hom(\hb^{i_1-n-1}((s_{i_1 j_1}-1)*\ub_{j_1}),\ub_{j_1}\hb^{i_1-n-s_{i_1 j_1}+1}) \\
&\qquad +\hb\Hom(\ub_{j_1}\hb^{i_1-n-s_{i_1 j_1}+1}, \hb^{i_1-n-1}((s_{i_1 j_1}-1)*\ub_{j_1})) \\
&= \hb^{2-s_{i_1 j_1}}\Hom(s_{i_1, j_1}*\ub_{j_1} - \ub_{j_1}\hb^{-s_{i_1 j_1}+1}, \ub_{j_1}) \\
&\qquad +\hb^{s_{i_1 j_1}-1}\Hom(\ub_{j_1}, s_{i_1 j_1}*\ub_{j_1}-\ub_{j_1}\hb^{-s_{i_1 j_1}+1}) \\
&= \hb^{2-s_{i_1 j_1}}\Hom(s_{i_1, j_1}*\ub_{j_1}, \ub_{j_1}) \\
&\qquad +\hb^{s_{i_1 j_1}-1}\Hom(\ub_{j_1}, s_{i_1 j_1}*\ub_{j_1}) \\
&\qquad -1-\hb.
\end{align*}

It is easily verified that $\Delta^{(j_1)}P_{2 j} = 0$ for all $j$. We have $\Delta^{(j_1)}P_{3 j}=0$ for $j\neq j_1$, and

\begin{align*}
\Delta^{(j_1)}P_{3 j_1} &= \Hom(\ub_{j_1}, \ub_{j_1}\hb^{-s_{i_1 j_1}+1}) \\
&= \hb^{-s_{i_1 j_1}+1}.
\end{align*}

Finally, we have

\begin{align*}
\Delta^{(j_1)}P_4 &= -\Hom(\ub_{j_1}\hb^{-s_{i_1 j_1}+1}, s_{i_1 j_1}*\ub_{j_1}) -\Hom(s_{i_1 j_1}*\ub_{j_1}, \ub_{j_1}\hb^{-s_{i_1 j_1}+1}) \\
&\qquad +\Hom(\ub_{j_1}\hb^{-s_{i_1 j_1}+1},\ub_{j_1}\hb^{-s_{i_1 j_1}+1}) \\
&= -\hb^{s_{i_1 j_1}-1}\Hom(\ub_{j_1}, s_{i_1 j_1}*\ub_{j_1}) -\hb^{-s_{i_1 j_1}+1}\Hom(s_{i_1 j_1}*\ub_{j_1}, \ub_{j_1})+1.
\end{align*}

Adding everything together yields

\begin{align*}
\Delta^{(j_1)}T\bowvar(\brane) &= \hb^{2-s_{i_1 j_1}}\Hom(s_{i_1 j_1}*\ub_{j_1},\ub_{j_1}) - \hb^{-s_{i_1 j_1}+1}\Hom(s_{i_1 j_1}*\ub_{j_1}, \ub_{j_1}) + \hb^{-s_{i_1 j_1}+1}-\hb \\
&= \hb^{2-s_{i_1 j_1}}\Hom(s_{i_1 j_1}*\ub_{j_1}-\hb(s_{i_1 j_1}*\ub_{j_1}), \ub_{j_1}) + \hb^{-s_{i_1 j_1}+1}-\hb \\
&= \hb^{2-s_{i_1 j_1}}\Hom(\ub_{j_1}\hb^{-s_{i_1 j_1}+1}-\ub_{j_1}\hb, \ub_{j_1}) + \hb^{-s_{i_1 j_1}+1}-\hb \\
\end{align*}

As $\Delta T\bowvar(\brane) = \sum_{j_0} \Delta^{(j_0)}T\bowvar(\brane)$, we have

\[
\Delta T\bowvar(\brane) = \sum_{\substack{j_0\\(i_1, j_0, j_1)\text{ 01-pair}}} \frac{\ub_{j_0}}{\ub_{j_1}}\hb^{s_{i_1 j_1}-s_{i_1 j_0}}
+\frac{\ub_{j_1}}{\ub_{j_0}}\hb^{1+s_{i_1 j_0}-s_{i_1 j_1}}.
\]

By our assumptions, switching $M_{i_1 j_1}$ from 1 to 0 cannot create any 01-pairs. Moreover, the 01-pairs destroyed by the change are all of the form $(i_1, j_0, j_1)$. Thus, the inductive step is proven.

Suppose now that $\brane$ is not separated and $p\in\bowvar(\brane)^\torus$. By applying a sequence of Hanany-Witten transitions, the tie diagram for $p$ can be transformed into a separated diagram $\widetilde\brane$, representing a fixed point $\widetilde p\in\bowvar(\widetilde\brane)^{\widetilde\torus}$ on a separated bow variety. This changes the separating line, but not the BCT. Hence, we can immediately apply Theorem <ref> to obtain the tangent weights at $\widetilde p$. As discussed in Section <ref>, these differ from the tangent weights at $p$ only in the exponents of $\hb$. Starting from $\widetilde\brane$ and going to $\brane$, each time we move a D5 brane $U$ to the left, we must multiply $\ub$ by $\hb$.

Let $\brane$ be a brane diagram and $p\in\bowvar(\brane)^\torus$. Let $M$ be the corresponding BCT and $\sigma_j$ be the number of entries in the $j$th column of $M$ lying below the separating line. Then, the tangent weights at $p$ are given by

\[
T_p\bowvar(\brane) = \sum_{(i, j_0, j_1)\text{ 01-pair}} \left(
\frac{\ub_{j_0}}{\ub_{j_1}}\hb^{s_{ij_1}-s_{ij_0} + \sigma_{j_0}-\sigma_{j_1}} + \frac{\ub_{j_1}}{\ub_{j_0}}\hb^{s_{ij_0}-s_{ij_1} + \sigma_{j_1}-\sigma_{j_0} +1} \right).
\]

Observe that the tangent weights come in pairs of the form $w+\hb w^{-1}$. This is a manisfestation of the fact that bow varieties admit polarization bundles <cit.>. Theorem <ref> combined with the dimension formula <cit.> gives us a roundabout proof of the following purely combinatorial result.

Let $M$ be a BCT with margins $r, c$. Then, the number of 01-pairs in $M$ is

\[
\frac{1}{2}\left(
\sum_{j=1}^m \left( \underline c_j(\underline c_j  + 1) + \underline c_{j-1}(\underline c_{j-1} + 1) \right)
+ 2\sum_{i=1}^n \underline r_{i-1}\underline r_i
-2\sum_{j=1}^m \underline c_j^2 -2\sum_{i=1}^{n-1} \underline r_i^2
\right),
\]

where $\underline c_j=\sum_{j'=1}^j c_{j'}$, and $\underline r_i=\sum_{i'=1}^i r_{i'}$. In particular, this number depends only on the margins.

§ BUTTERFLY SURGERIES AND INVARIANT CURVES

In this section, we will develop a combinatorial representation of $\torus$-invariant curves involving surgery operations on pairs of butterflies (see Section <ref>). Our goal is to identify all $\torus$-invariant curves. We will accomplish this using an exhaustive approach. For each fixed point $p$ and tangent weight $w$ that appears with multiplicity $k$ at $p$, we will produce a $k$-dimensional pencil of invariant curves with tangent weight $w$ at $p$.

§.§ Torus Invariant Curves

Let us begin by collecting some standard facts regarding invariant curves that will be used throughout what follows. These facts can be found in <cit.> and <cit.>. We address specifically the properties of invariant curves in bow varieties, though these properties are quite general and hold for a much larger family of spaces. Fix a bow variety $\bowvar(\brane)$ with the corresponding action by the torus $\torus$. By a “$\torus$-invariant curve" $\gamma$, we mean the closure of a 1-dimensional $\torus$-orbit in $\bowvar(\brane)$. The following are true of $\torus$-invariant curves:

* A 1-dimensional $\torus$-orbit is isomorphic to $\CC^\times$, and its closure is obtained by adding up to two $\torus$-fixed points.

* An invariant curve is compact if it contains two fixed points and noncompact otherwise. Compact invariant curves can be parametrized by equivariant embeddings $\CCP^1 \to \bowvar(\brane)$, and noncompact invariant curves with one fixed point can be parametrized by equivariant embeddings $\CC \to \bowvar(\brane)$.

* If $\gamma$ is an invariant curve, and $p \in \gamma$ is a fixed point, then $T_p \gamma$ is a $\torus$-weight space of $T_p \bowvar(\brane)$. If $p_1, p_2 \in \gamma$ are two distinct fixed points, then the $\torus$-weights of $T_{p_1}\gamma$ and $T_{p_2}\gamma$ are reciprocols.

* In contrast to [4], bow varieties may contain infinitely many invariant curves. In this case, the curves can be arranged into parametrized families or “pencils". Pencils of invariant curves will be described in Section <ref>. An invariant curve is a special case of a pencil, where the dimension of the pencil is 1.

The pencils of invariant curves have important implications for the study of the equivariant cohomology of $\bowvar(\brane)$. When $\bowvar(\brane)$ is equivariantly formal and has only finitely many fixed points and invariant curves (no multidimensional pencils), [4] shows that the map $H^*_\torus (\bowvar(\brane)) \to H^*_\torus (\bowvar(\brane)^\torus)$ induced by the inclusion of the fixed point locus is an injective ring homomorphism whose image is characterized by simple matching conditions on pairs of fixed points joined by an invariant curve. When multidimensional pencils of invariant curves are present, however, these matching conditions are not sufficient, and must be supplemented by more general matching conditions. See [3] and <cit.>. It is also appears that not all bow varieties are equivariantly formal <cit.>.

From now on, we will only be interested in invariant curves containing at least one fixed point. Is this automatically true if $\bowvar(\brane)$ has at least one fixed point? Floating copies of $\mathbb{C}^*$ with no fixed points. Can this happen? The following result will be pivotal in classifying all such curves.

Suppose $p$ is a $\torus$-fixed point contained in $\torus$-invariant curves $\gamma_1$ and $\gamma_2$. If $T_p\gamma_1=T_p\gamma_2$ as vector subspaces of $T_p\bowvar(\brane)$, then $\gamma_1 = \gamma_2$.

Let $p\in U \subset \bowvar(\brane)$ be an affine $\torus$-invariant neighborhood. By the Luna Slice Theorem, there exists a $\torus$-equivariant morphism $\phi:V \to T_p \bowvar(\brane)$ such that $p \in V \subset U$ is a $\torus$-invariant neighborhood, $\phi(p) = 0$, and $\phi$ is étale at $p$. Since our spaces are smooth, this étale morphism restricts to a diffeomorphism on some analytic neighborhood of $p$. The result now follows from the fact that the only invariant curves in $T_p\bowvar(\brane)$ containing $0$ are lines, which can be easily seen from the structure of the tangent weights provided by Corollary <ref>.

Our approach will be to construct a collection of invariant curves containing $p$ such that every $\torus$-invariant tangent line at $p$ appears as the tangent line to one of the curves. Then, Proposition <ref> will guarantee that all curves containing $p$ are accounted for.

§.§ Butterfly Surgery

[shift = (0, 21)]

[fill] (-1,-1) circle [radius = 0.025] node(0a1);

[fill] (0,0) circle [radius = 0.025] node(1a1); [fill] (0,-1) circle [radius = 0.025] node(1a2);

[fill] (1,1) circle [radius = 0.025] node(2a1); [fill] (1,0) circle [radius = 0.025] node(2a2); [fill] (1,-1) circle [radius = 0.025] node(2a3);

[fill] (2,2) circle [radius = 0.025] node(3a1); [fill] (2,1) circle [radius = 0.025] node(3a2); [fill] (2,0) circle [radius = 0.025] node(3a3);

[fill] (3,3) circle [radius = 0.025] node(4a1); [fill] (3,2) circle [radius = 0.025] node(4a2); [fill] (3,1) circle [radius = 0.025] node(4a3);
[fill] (3,0) circle [radius = 0.025] node(4a4);

[fill] (4,4) circle [radius = 0.025] node(5a1); [fill] (4,3) circle [radius = 0.025] node(5a2); [fill] (4,2) circle [radius = 0.025] node(5a3);
[fill] (4,1) circle [radius = 0.025] node(5a4);

[fill] (5,4) circle [radius = 0.025] node(6a1); [fill] (5,3) circle [radius = 0.025] node(6a2); [fill] (5,2) circle [radius = 0.025] node(6a3);
[fill] (5,1) circle [radius = 0.025] node(6a4);

[fill] (6,4) circle [radius = 0.025] node(7a1); [fill] (6,3) circle [radius = 0.025] node(7a2); [fill] (6,2) circle [radius = 0.025] node(7a3);
[fill] (6,1) circle [radius = 0.025] node(7a4);

[fill] (7,4) circle [radius = 0.025] node(8a1); [fill] (7,3) circle [radius = 0.025] node(8a2); [fill] (7,2) circle [radius = 0.025] node(8a3);
[fill] (7,1) circle [radius = 0.025] node(8a4);

[fill] (8,4) circle [radius = 0.025] node(9a1); [fill] (8,3) circle [radius = 0.025] node(9a2); [fill] (8,2) circle [radius = 0.025] node(9a3);
[fill] (8,1) circle [radius = 0.025] node(9a4);

[fill] (9,4) circle [radius = 0.025] node(10a1); [fill] (9,3) circle [radius = 0.025] node(10a2);
[fill] (9,2) circle [radius = 0.025] node(10a3);

[fill] (10,4) circle [radius = 0.025] node(11a1); [fill] (10,3) circle [radius = 0.025] node(11a2);

[fill] (11,4) circle [radius = 0.025] node(12a1); [fill] (11,3) circle [radius = 0.025] node(12a2);

[fill] (12,4) circle [radius = 0.025] node(13a1);

(4.5, 5) circle [radius = 0.05] node(a);

[->, red] (0a1)–(1a2);
[->, dotted, magenta] (1a1)–(0a1);

[->, red] (1a1)–(2a2); [->, red] (1a2)–(2a3);
[->, dotted, magenta] (2a1)–(1a1); [->, dotted, magenta] (2a2)–(1a2);

[->, red] (2a1)–(3a2); [->, red] (2a2)–(3a3);
[->, dotted, magenta] (3a2)–(2a2); [->, dotted, magenta] (3a1)–(2a1); [->, dotted, magenta] (3a3)–(2a3);

[->, red] (3a1)–(4a2); [->, red] (3a2)–(4a3); [->, red] (3a3)–(4a4);
[->, dotted, magenta] (4a1)–(3a1); [->, dotted, magenta] (4a2)–(3a2); [->, dotted, magenta] (4a3)–(3a3);

[->, red] (4a1)–(5a2); [->, red] (4a2)–(5a3); [->, red] (4a3)–(5a4);
[->, dotted, magenta] (5a1)–(4a1); [->, dotted, magenta] (5a2)–(4a2); [->, dotted, magenta] (5a3)–(4a3);
[->, dotted, magenta] (5a4)–(4a4);

[->, blue] (6a1)–(5a1); [->, blue] (6a2)–(5a2); [->, blue] (6a3)–(5a3); [->, blue] (6a4)–(5a4);
[->] (5a1)–(5a2); [->] (5a2)–(5a3); [->] (5a3)–(5a4); [->] (6a1)–(6a2); [->] (6a2)–(6a3); [->] (6a3)–(6a4);

[->, red] (6a1)–(7a1); [->, red] (6a2)–(7a2); [->, red] (6a3)–(7a3); [->, red] (6a4)–(7a4);
[->, dotted, magenta] (7a1)–(6a2); [->, dotted, magenta] (7a2)–(6a3); [->, dotted, magenta] (7a3)–(6a4);

[->, red] (7a1)–(8a1); [->, red] (7a2)–(8a2); [->, red] (7a3)–(8a3); [->, red] (7a4)–(8a4);
[->, dotted, magenta] (8a1)–(7a2); [->, dotted, magenta] (8a2)–(7a3); [->, dotted, magenta] (8a3)–(7a4);

[->, blue] (9a1)–(8a1); [->, blue] (9a2)–(8a2); [->, blue] (9a3)–(8a3); [->, blue] (9a4)–(8a4);
[->] (8a1)–(8a2); [->] (8a2)–(8a3); [->] (8a3)–(8a4); [->] (9a1)–(9a2); [->] (9a2)–(9a3); [->] (9a3)–(9a4);

[->, red] (9a1)–(10a1); [->, red] (9a2)–(10a2); [->, red] (9a3)–(10a3);
[->, dotted, magenta] (10a1)–(9a2); [->, dotted, magenta] (10a2)–(9a3); [->, dotted, magenta] (10a3)–(9a4);

[->, red] (10a1)–(11a1); [->, red] (10a2)–(11a2);
[->, dotted, magenta] (11a1)–(10a2); [->, dotted, magenta] (11a2)–(10a3);

[->, red] (11a1)–(12a1); [->, red] (11a2)–(12a2);
[->, dotted, magenta] (12a1)–(11a2);

[->, red] (12a1)–(13a1);
[->, dotted, magenta] (13a1)–(12a2);

[->, green] (a) to [in = 15, out = 245] (5a1);

[shift = (0, 14)]

[fill] (1,1) circle [radius = 0.025] node(2b1);

[fill] (2,2) circle [radius = 0.025] node(3b1);

[fill] (3,3) circle [radius = 0.025] node(4b1);

[fill] (4,4) circle [radius = 0.025] node(5b1); [fill] (4,3) circle [radius = 0.025] node(5b2);

[fill] (5,4) circle [radius = 0.025] node(6b1); [fill] (5,3) circle [radius = 0.025] node(6b2);

[fill] (6,5) circle [radius = 0.025] node(7b1); [fill] (6,4) circle [radius = 0.025] node(7b2);

[fill] (7,6) circle [radius = 0.025] node(8b1); [fill] (7,5) circle [radius = 0.025] node(8b2);

[fill] (8,5) circle [radius = 0.025] node(9b1);

[fill] (9,5) circle [radius = 0.025] node(10b1);

[fill] (10,5) circle [radius = 0.025] node(11b1);

[fill] (11,5) circle [radius = 0.025] node(12b1);

[fill] (12,5) circle [radius = 0.025] node(13b1);

(7.5, 7) circle [radius = 0.05] node(b);

[->, dotted, magenta] (3b1)–(2b1);

[->, dotted, magenta] (4b1)–(3b1);

[->, red] (4b1)–(5b2);
[->, dotted, magenta] (5b1)–(4b1);

[->, blue] (6b1)–(5b1); [->, blue] (6b2)–(5b2);
[->] (5b1)–(5b2); [->] (6b1)–(6b2);

[->, red] (6b1)–(7b2);
[->, dotted, magenta] (7b1)–(6b1); [->, dotted, magenta] (7b2)–(6b2);

[->, red] (7b1)–(8b2);
[->, dotted, magenta] (8b1)–(7b1); [->, dotted, magenta] (8b2)–(7b2);

[->, blue] (9b1)–(8b2);
[->] (8b1)–(8b2);

[->, red] (9b1)–(10b1);

[->, red] (10b1)–(11b1);

[->, red] (11b1)–(12b1);

[->, red] (12b1)–(13b1);

[->, green] (b) to [in = 15, out = 245] (8b1);

[shift = (0, 20)]
[gray] (-0.1, -0.1)–(1.05, -0.1)–(2.05, 0.9)–(3.05, 0.9)–(4.05, 1.9)–(8.05, 1.9)–(10.05, 3.9)–(11.1, 3.9)–(11.1, 4.1)–(9.95, 4.1)–(6.95, 4.1)–(4.95, 2.1)–(2.95, 2.1)–(0.95, 0.1)–(-0.1, 0.1)–(-0.1, -0.1);

[shift = (0, 14)]
[gray] (-0.1, -0.1)–(1.05, -0.1)–(2.05, 0.9)–(3.05, 0.9)–(4.05, 1.9)–(8.05, 1.9)–(10.05, 3.9)–(11.1, 3.9)–(11.1, 4.1)–(9.95, 4.1)–(6.95, 4.1)–(4.95, 2.1)–(2.95, 2.1)–(0.95, 0.1)–(-0.1, 0.1)–(-0.1, -0.1);

Fixed point with the site of a butterfly surgery outlined.

A fixed point $p_1\in \bowvar(\brane)^\torus$ corresponds to a butterfly diagram, that is, a collection of butterflies, one centered on each D5 brane. We stack the butterflies on top of each other, so that the centers of the butterflies listed from top to bottom go from left to right in $\brane$. Fix two butterflies $\but_1$ and $\but'_1$ with centers $U$ and $U'$, respectively. Suppose that there is a (possibly disconnected) subgraph $\site$ of $\but_1$, such that stacking $\site$ below $\but'_1$ and creating new edges between $\but'_1$ and $\site$ according to the rules of Section <ref> results in a butterfly $\but'_2$. The subgraph $\site$, called the site of the surgery, must be translated vertically, in a rigid fashion, without any lateral movement, rotation, or deformation, and we do not create new edges within $\site$. Let $\but_2$ be obtained by deleting $\site$ and all adjacent edges from $\but_1$, and assume that $\but_2$ is also a butterfly. The operation of replacing $\but_1$ with $\but_2$ and $\but'_1$ with $\but'_2$ is called a “butterfly surgery". It transforms the butterfly diagram for $p_1$ into the butterfly diagram for another fixed point $p_2\in \bowvar(\brane)^\torus$. We will use $\surgery$ to denote butterfly surgeries. Clearly, in a butterfly surgery, $\site$ must contain all edges of $\but_1$ with both endpoints within $\site$. Moreover, the 0-momentum condition forces $\site$ to be $A, A^{-1}, B, C, D$-invariant. By this, we mean that all $B, C, D$ edges in $\but_1$ incident to $\site$ are directed into $\site$, and $\site$ contains all incident $A$ edges.

An invariant curve arises from a butterfly surgery as follows. Consider the newly created edges in the butterfly diagram for $p_2$. Each edge starts in $\but'_1$ and ends in the translated copy of $\site$, which we will call $\site'$. Create the corresponding edges starting in $\but'_1$ and ending in $\site\subset\but_1$ in the butterfly diagram for $p_1$, as in Figure <ref>. Identifying vertices with basis vectors of the $W_M$ spaces and interpreting the edges as linear maps, the resulting graph gives an element $p_\gamma \in \prequot$. The 0-momentum condition holds by construction, as the rules in Section <ref> governing the vertices and edges of a butterfly directly reflect the 0-momentum conditions (see the proof of <cit.>). The S1, S2, and $\nu$ stability conditions hold, because adding arrows in this way expands the images and shrinks the kernels of the relevant maps.

[shift = (0, 7)]

[fill] (-1,-1) circle [radius = 0.025] node(0a1);

[fill] (0,0) circle [radius = 0.025] node(1a1);

[fill] (1,1) circle [radius = 0.025] node(2a1); [fill] (1,0) circle [radius = 0.025] node(2a2);

[fill] (2,2) circle [radius = 0.025] node(3a1); [fill] (2,1) circle [radius = 0.025] node(3a2);

[fill] (3,3) circle [radius = 0.025] node(4a1); [fill] (3,2) circle [radius = 0.025] node(4a2);

[fill] (4,4) circle [radius = 0.025] node(5a1); [fill] (4,3) circle [radius = 0.025] node(5a2); [fill] (4,2) circle [radius = 0.025] node(5a3);

[fill] (5,4) circle [radius = 0.025] node(6a1); [fill] (5,3) circle [radius = 0.025] node(6a2); [fill] (5,2) circle [radius = 0.025] node(6a3);

[fill] (6,4) circle [radius = 0.025] node(7a1); [fill] (6,3) circle [radius = 0.025] node(7a2);

[fill] (7,4) circle [radius = 0.025] node(8a1);

[fill] (8,4) circle [radius = 0.025] node(9a1);

[fill] (9,4) circle [radius = 0.025] node(10a1);

[fill] (10,4) circle [radius = 0.025] node(11a1);

[fill] (11,4) circle [radius = 0.025] node(12a1);

[fill] (12,4) circle [radius = 0.025] node(13a1);

(4.5, 5) circle [radius = 0.05] node(a);

[->, dotted, magenta] (1a1)–(0a1);

[->, red] (1a1)–(2a2); [->, dotted, magenta] (2a1)–(1a1);

[->, red] (2a1)–(3a2);
[->, dotted, magenta] (3a2)–(2a2); [->, dotted, magenta] (3a1)–(2a1);

[->, red] (3a1)–(4a2);
[->, dotted, magenta] (4a1)–(3a1); [->, dotted, magenta] (4a2)–(3a2);

[->, red] (4a1)–(5a2); [->, red] (4a2)–(5a3);
[->, dotted, magenta] (5a1)–(4a1); [->, dotted, magenta] (5a2)–(4a2);

[->, blue] (6a1)–(5a1); [->, blue] (6a2)–(5a2); [->, blue] (6a3)–(5a3);
[->] (5a1)–(5a2); [->] (5a2)–(5a3); [->] (6a1)–(6a2); [->] (6a2)–(6a3);

[->, red] (6a1)–(7a1); [->, red] (6a2)–(7a2);
[->, dotted, magenta] (7a1)–(6a2); [->, dotted, magenta] (7a2)–(6a3);

[->, red] (7a1)–(8a1);
[->, dotted, magenta] (8a1)–(7a2);

[->, blue] (9a1)–(8a1);

[->, red] (9a1)–(10a1);

[->, red] (10a1)–(11a1);

[->, red] (11a1)–(12a1);

[->, red] (12a1)–(13a1);

[->, green] (a) to [in = 15, out = 245] (5a1);

[fill] (0,0) circle [radius = 0.025] node(1a1);

[fill] (1,1) circle [radius = 0.025] node(2a1); [fill] (1,0) circle [radius = 0.025] node(2a2);

[fill] (2,2) circle [radius = 0.025] node(3a1); [fill] (2,1) circle [radius = 0.025] node(3a2);

[fill] (3,3) circle [radius = 0.025] node(4a1); [fill] (3,2) circle [radius = 0.025] node(4a2); [fill] (3,1) circle [radius = 0.025] node(4a3);

[fill] (4,4) circle [radius = 0.025] node(5a1); [fill] (4,3) circle [radius = 0.025] node(5a2); [fill] (4,2) circle [radius = 0.025] node(5a3);

[fill] (5,4) circle [radius = 0.025] node(6a1); [fill] (5,3) circle [radius = 0.025] node(6a2); [fill] (5,2) circle [radius = 0.025] node(6a3);

[fill] (6,5) circle [radius = 0.025] node(7a1); [fill] (6,4) circle [radius = 0.025] node(7a2); [fill] (6,3) circle [radius = 0.025] node(7a3);
[fill] (6,2) circle [radius = 0.025] node(7a4);

[fill] (7,6) circle [radius = 0.025] node(8a1); [fill] (7,5) circle [radius = 0.025] node(8a2); [fill] (7,4) circle [radius = 0.025] node(8a3);
[fill] (7,3) circle [radius = 0.025] node(8a4); [fill] (7,2) circle [radius = 0.025] node(8a5);

[fill] (8,5) circle [radius = 0.025] node(9a1); [fill] (8,4) circle [radius = 0.025] node(9a2); [fill] (8,3) circle [radius = 0.025] node(9a3);
[fill] (8,2) circle [radius = 0.025] node(9a4);

[fill] (9,5) circle [radius = 0.025] node(10a1); [fill] (9,4) circle [radius = 0.025] node(10a2);
[fill] (9,3) circle [radius = 0.025] node(10a3);

[fill] (10,5) circle [radius = 0.025] node(11a1); [fill] (10,4) circle [radius = 0.025] node(11a2);

[fill] (11,5) circle [radius = 0.025] node(12a1); [fill] (11,4) circle [radius = 0.025] node(12a2);

[fill] (12,5) circle [radius = 0.025] node(13a1);

(7.5, 7) circle [radius = 0.05] node(a);

[->, red] (1a1)–(2a2); [->, dotted, magenta] (2a1)–(1a1);

[->, red] (2a1)–(3a2);
[->, dotted, magenta] (3a2)–(2a2); [->, dotted, magenta] (3a1)–(2a1);

[->, red] (3a1)–(4a2); [->, red] (3a2)–(4a3);
[->, dotted, magenta] (4a1)–(3a1); [->, dotted, magenta] (4a2)–(3a2);

[->, red] (4a1)–(5a2); [->, red] (4a2)–(5a3);
[->, dotted, magenta] (5a1)–(4a1); [->, dotted, magenta] (5a2)–(4a2); [->, dotted, magenta] (5a3)–(4a3);

[->, blue] (6a1)–(5a1); [->, blue] (6a2)–(5a2); [->, blue] (6a3)–(5a3);
[->] (5a1)–(5a2); [->] (5a2)–(5a3); [->] (6a1)–(6a2); [->] (6a2)–(6a3);

[->, red] (6a1)–(7a2); [->, red] (6a2)–(7a3); [->, red] (6a3)–(7a4);
[->, dotted, magenta] (7a1)–(6a1); [->, dotted, magenta] (7a2)–(6a2); [->, dotted, magenta] (7a3)–(6a3);

[->, red] (7a1)–(8a2); [->, red] (7a2)–(8a3); [->, red] (7a3)–(8a4); [->, red] (7a4)–(8a5);
[->, dotted, magenta] (8a1)–(7a1); [->, dotted, magenta] (8a2)–(7a2); [->, dotted, magenta] (8a3)–(7a3);
[->, dotted, magenta] (8a4)–(7a4);

[->, blue] (9a1)–(8a2); [->, blue] (9a2)–(8a3); [->, blue] (9a3)–(8a4); [->, blue] (9a4)–(8a5);
[->] (8a1)–(8a2); [->] (8a2)–(8a3); [->] (8a3)–(8a4); [->] (8a4)–(8a5); [->] (9a1)–(9a2); [->] (9a2)–(9a3);
[->] (9a3)–(9a4);

[->, red] (9a1)–(10a1); [->, red] (9a2)–(10a2); [->, red] (9a3)–(10a3);
[->, dotted, magenta] (10a1)–(9a2); [->, dotted, magenta] (10a2)–(9a3); [->, dotted, magenta] (10a3)–(9a4);

[->, red] (10a1)–(11a1); [->, red] (10a2)–(11a2);
[->, dotted, magenta] (11a1)–(10a2); [->, dotted, magenta] (11a2)–(10a3);

[->, red] (11a1)–(12a1); [->, red] (11a2)–(12a2);
[->, dotted, magenta] (12a1)–(11a2);

[->, red] (12a1)–(13a1);
[->, dotted, magenta] (13a1)–(12a2);

[->, green] (a) to [in = 15, out = 245] (8a1);

[shift = (0, 6)]
[gray] (-0.1, -0.1)–(1.05, -0.1)–(2.05, 0.9)–(3.05, 0.9)–(4.05, 1.9)–(8.05, 1.9)–(10.05, 3.9)–(11.1, 3.9)–(11.1, 4.1)–(9.95, 4.1)–(6.95, 4.1)–(4.95, 2.1)–(2.95, 2.1)–(0.95, 0.1)–(-0.1, 0.1)–(-0.1, -0.1);

[gray] (-0.1, -0.1)–(1.05, -0.1)–(2.05, 0.9)–(3.05, 0.9)–(4.05, 1.9)–(8.05, 1.9)–(10.05, 3.9)–(11.1, 3.9)–(11.1, 4.1)–(9.95, 4.1)–(6.95, 4.1)–(4.95, 2.1)–(2.95, 2.1)–(0.95, 0.1)–(-0.1, 0.1)–(-0.1, -0.1);

New fixed point resulting from butterfly surgery.

For a vertex $v$ of the butterfly diagram, recall that its height (Definition <ref>) is denoted by $y(v)$. Now, let $v\in\site \subset \beta$. Applying the surgery yields the corresponding vertex $v'\in\site' \subset \beta'$. The change in height $\Delta_\surgery y = y(v') - y(v)$ is constant with respect to $v\in\site$. Consider the action of $\torus$ on $p_\gamma$. By a suitable change of basis in $W_X$ spaces (the $\G$-action), a representation of the resulting point can be obtained by multiplying the new edges by the weight $\hb^{-\Delta_\surgery y} \ub / \ub'$ and leaving the other edges alone. Denote the $\torus$-orbit closure of $p_\gamma$ by $\gamma$. By fixing $\ub, \hb$ and taking $\ub' \to \infty$, we see that $p_1 \in \gamma$. Moreover, there is a surjective map $\CC^\times \to \torus.p_\gamma$ given by sending $t$ to the point obtained by multiplying the new edges by $t$. It follows that $\gamma$ is at most 1-dimensional. We must rule out the possibility that $\gamma = p_1$. Note that the addition of the arrows has increased the ranks of some $B, C, D$ maps. Ranks are preserved by the $\G$ action, so $\gamma \neq p_1$. It follows that $\gamma$ is an invariant curve containing $p_1$. Moreover, its tangent weight at $p_1$ is precisely $\hb^{-\Delta_\surgery y} \ub / \ub'$.

[shift = (0, 21)]

[fill] (-1,-1) circle [radius = 0.025] node(0a1);

[fill] (0,0) circle [radius = 0.025] node(1a1); [fill] (0,-1) circle [radius = 0.025] node(1a2);

[fill] (1,1) circle [radius = 0.025] node(2a1); [fill] (1,0) circle [radius = 0.025] node(2a2); [fill] (1,-1) circle [radius = 0.025] node(2a3);

[fill] (2,2) circle [radius = 0.025] node(3a1); [fill] (2,1) circle [radius = 0.025] node(3a2); [fill] (2,0) circle [radius = 0.025] node(3a3);

[fill] (3,3) circle [radius = 0.025] node(4a1); [fill] (3,2) circle [radius = 0.025] node(4a2); [fill] (3,1) circle [radius = 0.025] node(4a3);
[fill] (3,0) circle [radius = 0.025] node(4a4);

[fill] (4,4) circle [radius = 0.025] node(5a1); [fill] (4,3) circle [radius = 0.025] node(5a2); [fill] (4,2) circle [radius = 0.025] node(5a3);
[fill] (4,1) circle [radius = 0.025] node(5a4);

[fill] (5,4) circle [radius = 0.025] node(6a1); [fill] (5,3) circle [radius = 0.025] node(6a2); [fill] (5,2) circle [radius = 0.025] node(6a3);
[fill] (5,1) circle [radius = 0.025] node(6a4);

[fill] (6,4) circle [radius = 0.025] node(7a1); [fill] (6,3) circle [radius = 0.025] node(7a2); [fill] (6,2) circle [radius = 0.025] node(7a3);
[fill] (6,1) circle [radius = 0.025] node(7a4);

[fill] (7,4) circle [radius = 0.025] node(8a1); [fill] (7,3) circle [radius = 0.025] node(8a2); [fill] (7,2) circle [radius = 0.025] node(8a3);
[fill] (7,1) circle [radius = 0.025] node(8a4);

[fill] (8,4) circle [radius = 0.025] node(9a1); [fill] (8,3) circle [radius = 0.025] node(9a2); [fill] (8,2) circle [radius = 0.025] node(9a3);
[fill] (8,1) circle [radius = 0.025] node(9a4);

[fill] (9,4) circle [radius = 0.025] node(10a1); [fill] (9,3) circle [radius = 0.025] node(10a2);
[fill] (9,2) circle [radius = 0.025] node(10a3);

[fill] (10,4) circle [radius = 0.025] node(11a1); [fill] (10,3) circle [radius = 0.025] node(11a2);

[fill] (11,4) circle [radius = 0.025] node(12a1); [fill] (11,3) circle [radius = 0.025] node(12a2);

[fill] (12,4) circle [radius = 0.025] node(13a1);

(4.5, 5) circle [radius = 0.05] node(a);

[->, red] (0a1)–(1a2);
[->, dotted, magenta] (1a1)–(0a1);

[->, red] (1a1)–(2a2); [->, red] (1a2)–(2a3);
[->, dotted, magenta] (2a1)–(1a1); [->, dotted, magenta] (2a2)–(1a2);

[->, red] (2a1)–(3a2); [->, red] (2a2)–(3a3);
[->, dotted, magenta] (3a2)–(2a2); [->, dotted, magenta] (3a1)–(2a1); [->, dotted, magenta] (3a3)–(2a3);

[->, red] (3a1)–(4a2); [->, red] (3a2)–(4a3); [->, red] (3a3)–(4a4);
[->, dotted, magenta] (4a1)–(3a1); [->, dotted, magenta] (4a2)–(3a2); [->, dotted, magenta] (4a3)–(3a3);

[->, red] (4a1)–(5a2); [->, red] (4a2)–(5a3); [->, red] (4a3)–(5a4);
[->, dotted, magenta] (5a1)–(4a1); [->, dotted, magenta] (5a2)–(4a2); [->, dotted, magenta] (5a3)–(4a3);
[->, dotted, magenta] (5a4)–(4a4);

[->, blue] (6a1)–(5a1); [->, blue] (6a2)–(5a2); [->, blue] (6a3)–(5a3); [->, blue] (6a4)–(5a4);
[->] (5a1)–(5a2); [->] (5a2)–(5a3); [->] (5a3)–(5a4); [->] (6a1)–(6a2); [->] (6a2)–(6a3); [->] (6a3)–(6a4);

[->, red] (6a1)–(7a1); [->, red] (6a2)–(7a2); [->, red] (6a3)–(7a3); [->, red] (6a4)–(7a4);
[->, dotted, magenta] (7a1)–(6a2); [->, dotted, magenta] (7a2)–(6a3); [->, dotted, magenta] (7a3)–(6a4);

[->, red] (7a1)–(8a1); [->, red] (7a2)–(8a2); [->, red] (7a3)–(8a3); [->, red] (7a4)–(8a4);
[->, dotted, magenta] (8a1)–(7a2); [->, dotted, magenta] (8a2)–(7a3); [->, dotted, magenta] (8a3)–(7a4);

[->, blue] (9a1)–(8a1); [->, blue] (9a2)–(8a2); [->, blue] (9a3)–(8a3); [->, blue] (9a4)–(8a4);
[->] (8a1)–(8a2); [->] (8a2)–(8a3); [->] (8a3)–(8a4); [->] (9a1)–(9a2); [->] (9a2)–(9a3); [->] (9a3)–(9a4);

[->, red] (9a1)–(10a1); [->, red] (9a2)–(10a2); [->, red] (9a3)–(10a3);
[->, dotted, magenta] (10a1)–(9a2); [->, dotted, magenta] (10a2)–(9a3); [->, dotted, magenta] (10a3)–(9a4);

[->, red] (10a1)–(11a1); [->, red] (10a2)–(11a2);
[->, dotted, magenta] (11a1)–(10a2); [->, dotted, magenta] (11a2)–(10a3);

[->, red] (11a1)–(12a1); [->, red] (11a2)–(12a2);
[->, dotted, magenta] (12a1)–(11a2);

[->, red] (12a1)–(13a1);
[->, dotted, magenta] (13a1)–(12a2);

[->, green] (a) to [in = 15, out = 245] (5a1);

[shift = (0, 14)]

[fill] (1,1) circle [radius = 0.025] node(2b1);

[fill] (2,2) circle [radius = 0.025] node(3b1);

[fill] (3,3) circle [radius = 0.025] node(4b1);

[fill] (4,4) circle [radius = 0.025] node(5b1); [fill] (4,3) circle [radius = 0.025] node(5b2);

[fill] (5,4) circle [radius = 0.025] node(6b1); [fill] (5,3) circle [radius = 0.025] node(6b2);

[fill] (6,5) circle [radius = 0.025] node(7b1); [fill] (6,4) circle [radius = 0.025] node(7b2);

[fill] (7,6) circle [radius = 0.025] node(8b1); [fill] (7,5) circle [radius = 0.025] node(8b2);

[fill] (8,5) circle [radius = 0.025] node(9b1);

[fill] (9,5) circle [radius = 0.025] node(10b1);

[fill] (10,5) circle [radius = 0.025] node(11b1);

[fill] (11,5) circle [radius = 0.025] node(12b1);

[fill] (12,5) circle [radius = 0.025] node(13b1);

(7.5, 7) circle [radius = 0.05] node(b);

[->, dotted, magenta] (3b1)–(2b1);

[->, dotted, magenta] (4b1)–(3b1);

[->, red] (4b1)–(5b2);
[->, dotted, magenta] (5b1)–(4b1);

[->, blue] (6b1)–(5b1); [->, blue] (6b2)–(5b2);
[->] (5b1)–(5b2); [->] (6b1)–(6b2);

[->, red] (6b1)–(7b2);
[->, dotted, magenta] (7b1)–(6b1); [->, dotted, magenta] (7b2)–(6b2);

[->, red] (7b1)–(8b2);
[->, dotted, magenta] (8b1)–(7b1); [->, dotted, magenta] (8b2)–(7b2);

[->, blue] (9b1)–(8b2);
[->] (8b1)–(8b2);

[->, red] (9b1)–(10b1);

[->, red] (10b1)–(11b1);

[->, red] (11b1)–(12b1);

[->, red] (12b1)–(13b1);

[->, green] (b) to [in = 15, out = 245] (8b1);

[shift = (0, 20)]
[gray] (-0.1, -0.1)–(1.05, -0.1)–(2.05, 0.9)–(3.05, 0.9)–(4.05, 1.9)–(8.05, 1.9)–(10.05, 3.9)–(11.1, 3.9)–(11.1, 4.1)–(9.95, 4.1)–(6.95, 4.1)–(4.95, 2.1)–(2.95, 2.1)–(0.95, 0.1)–(-0.1, 0.1)–(-0.1, -0.1);

[shift = (0, 14)]
[gray] (-0.1, -0.1)–(1.05, -0.1)–(2.05, 0.9)–(3.05, 0.9)–(4.05, 1.9)–(8.05, 1.9)–(10.05, 3.9)–(11.1, 3.9)–(11.1, 4.1)–(9.95, 4.1)–(6.95, 4.1)–(4.95, 2.1)–(2.95, 2.1)–(0.95, 0.1)–(-0.1, 0.1)–(-0.1, -0.1);

[->, red, dashed] (2b1) to[out = 0, in = 180] (3a3); [->, magenta, dashed] (2b1) to[out = 225, in = 45] (1a2);

[->, red, dashed] (3b1) to[out = 0, in = 180] (4a3);

[->, magenta, dashed] (5b2) to[out = 225, in = 45] (4a3); [->, dashed] (5b2) to[out = 270, in = 290] (5a4);

[->, red, dashed] (6b2) to[out = 0, in = 180] (7a3); [->, dashed] (6b2) to[out = 270, in = 255] (6a4);

[->, red, dashed] (7b2) to[out = 0, in = 180] (8a2);

[->, dashed] (8b2) to[out = 270, in = 290] (8a2);

[->, dashed] (9b1) to[out = 270, in = 255] (9a2);

[->, magenta, dashed] (10b1) to[out = 225, in = 45] (9a2);

[->, magenta, dashed] (11b1) to[out = 225, in = 45] (10a2);

[->, magenta, dashed] (12b1) to[out = 225, in = 45] (11a2);

[->, magenta, dashed] (13b1) to[out = 225, in = 45] (12a2);

Explicit construction of an invariant curve corresponding to butterfly surgery. The dashed lines are the new edges added to the butterfly diagram. Taking the $\torus$-orbit closure of the element of $\prequot$ represented by this graph yields an invariant curve with tangent weight $\left(\frac{\ub_1}{\ub_2}\hb\right)^{\pm 1}$ at its fixed points.

Next, set $\ub' = \hb = 1$, so that the new edges are acted upon by $\ub$. Apply the $\G$-action that multiplies each vertex in $\site$ by $\ub^{-1}$. Since all edges in $\beta_1$ adjacent to $\site$ point into $\site$, those edges will be multiplied by $\ub^{-1}$. Meanwhile, the factor on the new edges will cancel to 1, and all other edges will be unchanged. Taking $\ub \to \infty$, the edges adjacent to $\site$ are killed, leaving behind $\beta_2$ and $\beta_2'$. Thus, $p_2\in\gamma$, and $\gamma$ is a compact curve. Its tangent weight at $p_2$ is $\hb^{\Delta_\surgery y} \ub' / \ub$.

[shift = (0, 21)]

[fill] (-1,-1) circle [radius = 0.025] node(0a1);

[fill] (0,0) circle [radius = 0.025] node(1a1); [fill] (0,-1) circle [radius = 0.025] node(1a2);

[fill] (1,1) circle [radius = 0.025] node(2a1); [fill] (1,0) circle [radius = 0.025] node(2a2); [fill] (1,-1) circle [radius = 0.025] node(2a3);

[fill] (2,2) circle [radius = 0.025] node(3a1); [fill] (2,1) circle [radius = 0.025] node(3a2); [fill] (2,0) circle [radius = 0.025] node(3a3);

[fill] (3,3) circle [radius = 0.025] node(4a1); [fill] (3,2) circle [radius = 0.025] node(4a2); [fill] (3,1) circle [radius = 0.025] node(4a3);
[fill] (3,0) circle [radius = 0.025] node(4a4);

[fill] (4,4) circle [radius = 0.025] node(5a1); [fill] (4,3) circle [radius = 0.025] node(5a2); [fill] (4,2) circle [radius = 0.025] node(5a3);
[fill] (4,1) circle [radius = 0.025] node(5a4);

[fill] (5,4) circle [radius = 0.025] node(6a1); [fill] (5,3) circle [radius = 0.025] node(6a2); [fill] (5,2) circle [radius = 0.025] node(6a3);
[fill] (5,1) circle [radius = 0.025] node(6a4);

[fill] (6,4) circle [radius = 0.025] node(7a1); [fill] (6,3) circle [radius = 0.025] node(7a2); [fill] (6,2) circle [radius = 0.025] node(7a3);
[fill] (6,1) circle [radius = 0.025] node(7a4);

[fill] (7,4) circle [radius = 0.025] node(8a1); [fill] (7,3) circle [radius = 0.025] node(8a2); [fill] (7,2) circle [radius = 0.025] node(8a3);
[fill] (7,1) circle [radius = 0.025] node(8a4);

[fill] (8,4) circle [radius = 0.025] node(9a1); [fill] (8,3) circle [radius = 0.025] node(9a2); [fill] (8,2) circle [radius = 0.025] node(9a3);
[fill] (8,1) circle [radius = 0.025] node(9a4);

[fill] (9,4) circle [radius = 0.025] node(10a1); [fill] (9,3) circle [radius = 0.025] node(10a2);
[fill] (9,2) circle [radius = 0.025] node(10a3);

[fill] (10,4) circle [radius = 0.025] node(11a1); [fill] (10,3) circle [radius = 0.025] node(11a2);

[fill] (11,4) circle [radius = 0.025] node(12a1); [fill] (11,3) circle [radius = 0.025] node(12a2);

[fill] (12,4) circle [radius = 0.025] node(13a1);

(4.5, 5) circle [radius = 0.05] node(a);

[->, red] (0a1)–(1a2);
[->, dotted, magenta] (1a1)–(0a1);

[->, red] (1a1)–(2a2); [->, red] (1a2)–(2a3);
[->, dotted, magenta] (2a1)–(1a1); [->, dotted, magenta] (2a2)–(1a2);

[->, red] (2a1)–(3a2); [->, red] (2a2)–(3a3);
[->, dotted, magenta] (3a2)–(2a2); [->, dotted, magenta] (3a1)–(2a1); [->, dotted, magenta] (3a3)–(2a3);

[->, red] (3a1)–(4a2); [->, red] (3a2)–(4a3); [->, red] (3a3)–(4a4);
[->, dotted, magenta] (4a1)–(3a1); [->, dotted, magenta] (4a2)–(3a2); [->, dotted, magenta] (4a3)–(3a3);

[->, red] (4a1)–(5a2); [->, red] (4a2)–(5a3); [->, red] (4a3)–(5a4);
[->, dotted, magenta] (5a1)–(4a1); [->, dotted, magenta] (5a2)–(4a2); [->, dotted, magenta] (5a3)–(4a3);
[->, dotted, magenta] (5a4)–(4a4);

[->, blue] (6a1)–(5a1); [->, blue] (6a2)–(5a2); [->, blue] (6a3)–(5a3); [->, blue] (6a4)–(5a4);
[->] (5a1)–(5a2); [->] (5a2)–(5a3); [->] (5a3)–(5a4); [->] (6a1)–(6a2); [->] (6a2)–(6a3); [->] (6a3)–(6a4);

[->, red] (6a1)–(7a1); [->, red] (6a2)–(7a2); [->, red] (6a3)–(7a3); [->, red] (6a4)–(7a4);
[->, dotted, magenta] (7a1)–(6a2); [->, dotted, magenta] (7a2)–(6a3); [->, dotted, magenta] (7a3)–(6a4);

[->, red] (7a1)–(8a1); [->, red] (7a2)–(8a2); [->, red] (7a3)–(8a3); [->, red] (7a4)–(8a4);
[->, dotted, magenta] (8a1)–(7a2); [->, dotted, magenta] (8a2)–(7a3); [->, dotted, magenta] (8a3)–(7a4);

[->, blue] (9a1)–(8a1); [->, blue] (9a2)–(8a2); [->, blue] (9a3)–(8a3); [->, blue] (9a4)–(8a4);
[->] (8a1)–(8a2); [->] (8a2)–(8a3); [->] (8a3)–(8a4); [->] (9a1)–(9a2); [->] (9a2)–(9a3); [->] (9a3)–(9a4);

[->, red] (9a1)–(10a1); [->, red] (9a2)–(10a2); [->, red] (9a3)–(10a3);
[->, dotted, magenta] (10a1)–(9a2); [->, dotted, magenta] (10a2)–(9a3); [->, dotted, magenta] (10a3)–(9a4);

[->, red] (10a1)–(11a1); [->, red] (10a2)–(11a2);
[->, dotted, magenta] (11a1)–(10a2); [->, dotted, magenta] (11a2)–(10a3);

[->, red] (11a1)–(12a1); [->, red] (11a2)–(12a2);
[->, dotted, magenta] (12a1)–(11a2);

[->, red] (12a1)–(13a1);
[->, dotted, magenta] (13a1)–(12a2);

[->, green] (a) to [in = 15, out = 245] (5a1);

[shift = (0, 14)]

[fill] (1,1) circle [radius = 0.025] node(2b1);

[fill] (2,2) circle [radius = 0.025] node(3b1);

[fill] (3,3) circle [radius = 0.025] node(4b1);

[fill] (4,4) circle [radius = 0.025] node(5b1); [fill] (4,3) circle [radius = 0.025] node(5b2);

[fill] (5,4) circle [radius = 0.025] node(6b1); [fill] (5,3) circle [radius = 0.025] node(6b2);

[fill] (6,5) circle [radius = 0.025] node(7b1); [fill] (6,4) circle [radius = 0.025] node(7b2);

[fill] (7,6) circle [radius = 0.025] node(8b1); [fill] (7,5) circle [radius = 0.025] node(8b2);

[fill] (8,5) circle [radius = 0.025] node(9b1);

[fill] (9,5) circle [radius = 0.025] node(10b1);

[fill] (10,5) circle [radius = 0.025] node(11b1);

[fill] (11,5) circle [radius = 0.025] node(12b1);

[fill] (12,5) circle [radius = 0.025] node(13b1);

(7.5, 7) circle [radius = 0.05] node(b);

[->, dotted, magenta] (3b1)–(2b1);

[->, dotted, magenta] (4b1)–(3b1);

[->, red] (4b1)–(5b2);
[->, dotted, magenta] (5b1)–(4b1);

[->, blue] (6b1)–(5b1); [->, blue] (6b2)–(5b2);
[->] (5b1)–(5b2); [->] (6b1)–(6b2);

[->, red] (6b1)–(7b2);
[->, dotted, magenta] (7b1)–(6b1); [->, dotted, magenta] (7b2)–(6b2);

[->, red] (7b1)–(8b2);
[->, dotted, magenta] (8b1)–(7b1); [->, dotted, magenta] (8b2)–(7b2);

[->, blue] (9b1)–(8b2);
[->] (8b1)–(8b2);

[->, red] (9b1)–(10b1);

[->, red] (10b1)–(11b1);

[->, red] (11b1)–(12b1);

[->, red] (12b1)–(13b1);

[->, green] (b) to [in = 15, out = 245] (8b1);

[gray] (4a4) +(-0.1,0.1)–+(0.1,0.1)–+(0.1,-0.1)–+(-0.1,-0.1)–+(-0.1,0.1);
[gray] (7a4) ++(-0.1,0) ++(-0.1,-0.1)–++(2.2,0)–++(1.2,1.2)–++(-2.2,0)–++(-1.2,-1.2);

[gray] (4a4) ++(0,-5) +(-0.1,0.1)–+(0.1,0.1)–+(0.1,-0.1)–+(-0.1,-0.1)–+(-0.1,0.1);
[gray] (7a4) ++(-0.1,-5) ++(-0.1,-0.1)–++(2.2,0)–++(1.2,1.2)–++(-2.2,0)–++(-1.2,-1.2);

(3a3) node at +(0,-2.5)$z_1$;
[red,dashed,->] (3b1) to[in=180,out=0] (4a4);
[magenta,dashed,->] (5b2) to[in=45,out=225] (4a4);

(6a4) node at +(0,-2.5)$z_2$;
[red,dashed,->] (6b2) to[in=180,out=0] (7a4);
[red,dashed,->] (7b2) to[in=180,out=0] (8a3);
[magenta,dashed,->] (10b1) to[in=45,out=225] (9a3);
[magenta,dashed,->] (11b1) to[in=45,out=225] (10a3);
[->, dashed] (8b2) to[out = 270, in = 290] (8a3);
[->, dashed] (9b1) to[out = 270, in = 255] (9a3);

Explicit 2-dimensional pencil of invariant curves with tangent weight $\left(\frac{\ub_1}{\ub_2}\right)^{\pm 1}$ at its fixed points.

From general considerations, $\gamma$ is isomorphic to $\CCP^1$. We obtain an explicit parametrization by multiplying each of the new edges by $t\in\CC$. This gives us an isomorphism $\CC \to \gamma \setminus\{p_2\}$. Compactifying $\CC$ with a point $\infty$ and mapping $\infty$ to $p_2$ yields a parametrized curve $\gamma:\CCP^1 \to M$ (we are abusing notation and using $\gamma$ to refer both to the parametrization and its image). Note that performing the inverse butterfly surgery yields the same invariant curve with a different parametrization, one with $\gamma(0) = p_2$.

Suppose $\site$ has $k$ connected components $\site_1,...,\site_k$. By multiplying the new edges pointing into $\site_i$ by $z_i$, where $z = (z_1, ..., z_k)\in\CC^k\setminus\{0\}$, we obtain a $k$-parameter family $\Gamma_\surgery$ of $\torus$-invariant curves containing $p_1$ and $p_2$. We may projectivize using the $\G$-action to obtain a map $\pen_\surgery:\CCP^{k-1}\times\CCP^1 \to \bowvar(\brane)$. The $\CCP^{k-1}$ factor picks out a specific curve of $\Gamma_\surgery$ and the $\CCP^1$ factor parametrizes that curve as described in the previous paragraph. Consider the $k$ butterfly surgeries corresponding to each of the $k$ connected components of $\site$. These butterfly surgeries produce invariant curves $\gamma_1, ..., \gamma_k$. Clearly, the tangent spaces of these curves at $p_1$ (and also at $p_2$) are linearly independent. It is also clear that

\[
\forall z\in\CC^k \setminus\{0\},\ \ \left. \frac{\partial}{\partial t} \right|_{t=0} \pen_\surgery([z], t) = \sum_{i = 1}^k z_i\gamma_i'(0).
\]

It follows that each invariant curve in $\Gamma_\surgery$ has a distinct tangent space at $p_1$ and is therefore distinct. In other words, $\pen_\surgery$ is injective on $\CCP^{k-1} \times \CC^\times \subset \CCP^{k-1}\times\CCP^1$. The results of this section are summarized in the following lemma.

Suppose a butterfly surgery $\surgery$ sends a $\torus$-fixed point $p_1$ to the $\torus$-fixed point $p_2$. Let the site $\site$ be moved from the $U$ butterfly to the $U'$ butterfly by $\surgery$. Then, there is a $k$-dimensional pencil $\pen_\surgery:\CCP^{k-1}\times \CCP^1\to\bowvar(\brane)$ of compact invariant curves containing $p_1$ and $p_2$, where $k$ is the number of connected components of $\site$. Moreover, each curve in the pencil has tangent weight

\[
\frac{\ub}{\ub'}\hb^{-\Delta_\surgery y},
\]

at $p_1$, where

\[
\Delta_\surgery y = y(v') - y(v),
\]

for any $v\in\site$ and corresponding $v'\in\site'$. Here, $y$ denotes the height of a vertex (Definition <ref>).

It will be shown that butterfly surgeries capture all of the compact invariant curves in $\bowvar(\brane)$. To do so, our approach also requires us to construct noncompact invariant curves.

§.§ Botched Butterfly Surgery

For the remainder of Section <ref>, we will assume that $M$ is separated. Analogous results will follow for the nonseparated case by tracing the relevant constructions through Hanany-Witten transitions. See the proof of <cit.> for the explicit construction of the Hanany-Witten isomorphism. We explicitly trace a butterfly through the Hanany-Witten transition in <cit.>. First, let us define a surgery operation that closely resembles a butterfly surgery, but is not exactly a butterfly surgery.

Let $\brane$ be a separated brane diagram with $n$ NS5 branes and $m$ D5 branes. As in Section <ref>, index the branes from left to right. The NS5 branes are denoted $V_1, ..., V_n$, the D5 branes are denoted $U_1, ..., U_m$, and the segments (D3 branes) are denoted $X_0, X_1, ..., X_{n+m}$, where $X_0$ and $X_{n+m}$ are the infinite left and right segments. We will divide $\brane$ into a “left side" and a “right side". The left side comprises $X_0, ..., X_n$ and $V_1, ..., V_n$. The right side is the remaining portion of the diagram. The notion of left and right side extends naturally to butterflies and butterfly diagrams. The left side of a butterfly always has a certain inverted staircase-like shape. See Figure <ref> for an example.

[dashed, cyan] (29, 2)–(29, -58);

[thick,red] (-0.5,-1)–(0.5,1) node[pos=0.5](V1) node[pos=1,inner sep=0](V1t) node[pos=0,inner sep=0](V1b);
[thick,red] (2.5,-1)–(3.5,1) node[pos=0.5](V2) node[pos=1,inner sep=0](V2t) node[pos=0,inner sep=0](V2b);
[thick,red] (5.5,-1)–(6.5,1) node[pos=0.5](V3) node[pos=1,inner sep=0](V3t) node[pos=0,inner sep=0](V3b);
[thick,red] (8.5,-1)–(9.5,1) node[pos=0.5](V4) node[pos=1,inner sep=0](V4t) node[pos=0,inner sep=0](V4b);
[thick,red] (11.5,-1)–(12.5,1) node[pos=0.5](V5) node[pos=1,inner sep=0](V5t) node[pos=0,inner sep=0](V5b);
[thick,red] (14.5,-1)–(15.5,1) node[pos=0.5](V6) node[pos=1,inner sep=0](V6t) node[pos=0,inner sep=0](V6b);
[thick,red] (17.5,-1)–(18.5,1) node[pos=0.5](V7) node[pos=1,inner sep=0](V7t) node[pos=0,inner sep=0](V7b);
[thick,red] (20.5,-1)–(21.5,1) node[pos=0.5](V8) node[pos=1,inner sep=0](V8t) node[pos=0,inner sep=0](V8b);
[thick,red] (23.5,-1)–(24.5,1) node[pos=0.5](V9) node[pos=1,inner sep=0](V9t) node[pos=0,inner sep=0](V9b);
[thick,red] (26.5,-1)–(27.5,1) node[pos=0.5](V10) node[pos=1,inner sep=0](V10t) node[pos=0,inner sep=0](V10b);
[thick,blue] (30.5,-1)–(29.5,1) node[pos=0.5](U1) node[pos=1,inner sep=0](U1t) node[pos=0,inner sep=0](U1b);
[thick,blue] (33.5,-1)–(32.5,1) node[pos=0.5](U2) node[pos=1,inner sep=0](U2t) node[pos=0,inner sep=0](U2b);
[thick,blue] (36.5,-1)–(35.5,1) node[pos=0.5](U3) node[pos=1,inner sep=0](U3t) node[pos=0,inner sep=0](U3b);

(V1)–(V2) node[pos=0.5, above](X1)1;
(V2)–(V3) node[pos=0.5, above](X2)2;
(V3)–(V4) node[pos=0.5, above](X3)3;
(V4)–(V5) node[pos=0.5, above](X4)4;
(V5)–(V6) node[pos=0.5, above](X5)5;
(V6)–(V7) node[pos=0.5, above](X6)7;
(V7)–(V8) node[pos=0.5, above](X7)9;
(V8)–(V9) node[pos=0.5, above](X8)9;
(V9)–(V10) node[pos=0.5, above](X9)10;
(V10)–(U1) node[pos=0.5, above](X10)11;
(U1)–(U2) node[pos=0.5, above](X11)6;
(U2)–(U3) node[pos=0.5, above](X12)4;

() at (0,-6) ;

(30,-3) circle[radius=0.14] node(u1);
[fill] (1.5,-33) circle[radius=0.07] node(u1a1a1);
[fill] (4.5,-30) circle[radius=0.07] node(u1a2a1);

[fill] (7.5,-27) circle[radius=0.07] node(u1a3a1);

[fill] (10.5,-24) circle[radius=0.07] node(u1a4a1);
[fill] (10.5,-27) circle[radius=0.07] node(u1a4a2);
[fill] (13.5,-21) circle[radius=0.07] node(u1a5a1);
[fill] (13.5,-24) circle[radius=0.07] node(u1a5a2);
[fill] (13.5,-27) circle[radius=0.07] node(u1a5a3);
[fill] (16.5,-18) circle[radius=0.07] node(u1a6a1);
[fill] (16.5,-21) circle[radius=0.07] node(u1a6a2);
[fill] (16.5,-24) circle[radius=0.07] node(u1a6a3);
[fill] (16.5,-27) circle[radius=0.07] node(u1a6a4);
[fill] (19.5,-15) circle[radius=0.07] node(u1a7a1);
[fill] (19.5,-18) circle[radius=0.07] node(u1a7a2);
[fill] (19.5,-21) circle[radius=0.07] node(u1a7a3);
[fill] (19.5,-24) circle[radius=0.07] node(u1a7a4);
[fill] (22.5,-12) circle[radius=0.07] node(u1a8a1);
[fill] (22.5,-15) circle[radius=0.07] node(u1a8a2);
[fill] (22.5,-18) circle[radius=0.07] node(u1a8a3);
[fill] (22.5,-21) circle[radius=0.07] node(u1a8a4);
[fill] (25.5,-9) circle[radius=0.07] node(u1a9a1);
[fill] (25.5,-12) circle[radius=0.07] node(u1a9a2);
[fill] (25.5,-15) circle[radius=0.07] node(u1a9a3);
[fill] (25.5,-18) circle[radius=0.07] node(u1a9a4);
[fill] (25.5,-21) circle[radius=0.07] node(u1a9a5);
[fill] (28.5,-6) circle[radius=0.07] node(u1a10a1);
[fill] (28.5,-9) circle[radius=0.07] node(u1a10a2);
[fill] (28.5,-12) circle[radius=0.07] node(u1a10a3);
[fill] (28.5,-15) circle[radius=0.07] node(u1a10a4);
[fill] (28.5,-18) circle[radius=0.07] node(u1a10a5);

[->, green] (u1) to[out=-120,in=70] (u1a10a1);
[->,dotted,magenta] (u1a2a1)–(u1a1a1);

[->,dotted,magenta] (u1a3a1)–(u1a2a1);

[->,dotted,magenta] (u1a4a1)–(u1a3a1);

[->,red] (u1a3a1)–(u1a4a2);
[->,dotted,magenta] (u1a5a1)–(u1a4a1);
[->,dotted,magenta] (u1a5a2)–(u1a4a2);
[->,red] (u1a4a1)–(u1a5a2);
[->,red] (u1a4a2)–(u1a5a3);
[->,dotted,magenta] (u1a6a1)–(u1a5a1);
[->,dotted,magenta] (u1a6a2)–(u1a5a2);
[->,dotted,magenta] (u1a6a3)–(u1a5a3);
[->,red] (u1a5a1)–(u1a6a2);
[->,red] (u1a5a2)–(u1a6a3);
[->,red] (u1a5a3)–(u1a6a4);
[->,dotted,magenta] (u1a7a1)–(u1a6a1);
[->,dotted,magenta] (u1a7a2)–(u1a6a2);
[->,dotted,magenta] (u1a7a3)–(u1a6a3);
[->,dotted,magenta] (u1a7a4)–(u1a6a4);
[->,red] (u1a6a1)–(u1a7a2);
[->,red] (u1a6a2)–(u1a7a3);
[->,red] (u1a6a3)–(u1a7a4);
[->,dotted,magenta] (u1a8a1)–(u1a7a1);
[->,dotted,magenta] (u1a8a2)–(u1a7a2);
[->,dotted,magenta] (u1a8a3)–(u1a7a3);
[->,dotted,magenta] (u1a8a4)–(u1a7a4);
[->,red] (u1a7a1)–(u1a8a2);
[->,red] (u1a7a2)–(u1a8a3);
[->,red] (u1a7a3)–(u1a8a4);
[->,dotted,magenta] (u1a9a1)–(u1a8a1);
[->,dotted,magenta] (u1a9a2)–(u1a8a2);
[->,dotted,magenta] (u1a9a3)–(u1a8a3);
[->,dotted,magenta] (u1a9a4)–(u1a8a4);
[->,red] (u1a8a1)–(u1a9a2);
[->,red] (u1a8a2)–(u1a9a3);
[->,red] (u1a8a3)–(u1a9a4);
[->,red] (u1a8a4)–(u1a9a5);
[->,dotted,magenta] (u1a10a1)–(u1a9a1);
[->,dotted,magenta] (u1a10a2)–(u1a9a2);
[->,dotted,magenta] (u1a10a3)–(u1a9a3);
[->,dotted,magenta] (u1a10a4)–(u1a9a4);
[->,dotted,magenta] (u1a10a5)–(u1a9a5);
[->,red] (u1a9a1)–(u1a10a2);
[->,red] (u1a9a2)–(u1a10a3);
[->,red] (u1a9a3)–(u1a10a4);
[->,red] (u1a9a4)–(u1a10a5);
[->] (u1a10a1)–(u1a10a2);
[->] (u1a10a2)–(u1a10a3);
[->] (u1a10a3)–(u1a10a4);
[->] (u1a10a4)–(u1a10a5);

() at (0,-33) ;

(33,-30) circle[radius=0.14] node(u2);
[fill] (16.5,-45) circle[radius=0.07] node(u2a6a1);
[fill] (19.5,-42) circle[radius=0.07] node(u2a7a1);
[fill] (19.5,-45) circle[radius=0.07] node(u2a7a2);
[fill] (22.5,-39) circle[radius=0.07] node(u2a8a1);
[fill] (22.5,-42) circle[radius=0.07] node(u2a8a2);
[fill] (25.5,-36) circle[radius=0.07] node(u2a9a1);
[fill] (25.5,-39) circle[radius=0.07] node(u2a9a2);
[fill] (28.5,-33) circle[radius=0.07] node(u2a10a1);
[fill] (28.5,-36) circle[radius=0.07] node(u2a10a2);
[fill] (31.5,-33) circle[radius=0.07] node(u2a11a1);
[fill] (31.5,-36) circle[radius=0.07] node(u2a11a2);

[->, green] (u2) to[out=-120,in=70] (u2a11a1);
[->,dotted,magenta] (u2a7a1)–(u2a6a1);
[->,red] (u2a6a1)–(u2a7a2);
[->,dotted,magenta] (u2a8a1)–(u2a7a1);
[->,dotted,magenta] (u2a8a2)–(u2a7a2);
[->,red] (u2a7a1)–(u2a8a2);
[->,dotted,magenta] (u2a9a1)–(u2a8a1);
[->,dotted,magenta] (u2a9a2)–(u2a8a2);
[->,red] (u2a8a1)–(u2a9a2);
[->,dotted,magenta] (u2a10a1)–(u2a9a1);
[->,dotted,magenta] (u2a10a2)–(u2a9a2);
[->,red] (u2a9a1)–(u2a10a2);
[->, blue] (u2a11a2)–(u2a10a2);
[->, blue] (u2a11a1)–(u2a10a1);
[->] (u2a10a1)–(u2a10a2);
[->] (u2a11a1)–(u2a11a2);

() at (0,-45) ;

(36,-42) circle[radius=0.14] node(u3);
[fill] (4.5,-69) circle[radius=0.07] node(u3a2a1);
[fill] (7.5,-66) circle[radius=0.07] node(u3a3a1);
[fill] (7.5,-69) circle[radius=0.07] node(u3a3a2);
[fill] (10.5,-63) circle[radius=0.07] node(u3a4a1);
[fill] (10.5,-66) circle[radius=0.07] node(u3a4a2);
[fill] (13.5,-60) circle[radius=0.07] node(u3a5a1);
[fill] (13.5,-63) circle[radius=0.07] node(u3a5a2);
[fill] (16.5,-57) circle[radius=0.07] node(u3a6a1);
[fill] (16.5,-60) circle[radius=0.07] node(u3a6a2);
[fill] (19.5,-54) circle[radius=0.07] node(u3a7a1);
[fill] (19.5,-57) circle[radius=0.07] node(u3a7a2);
[fill] (19.5,-60) circle[radius=0.07] node(u3a7a3);
[fill] (22.5,-51) circle[radius=0.07] node(u3a8a1);
[fill] (22.5,-54) circle[radius=0.07] node(u3a8a2);
[fill] (22.5,-57) circle[radius=0.07] node(u3a8a3);
[fill] (25.5,-48) circle[radius=0.07] node(u3a9a1);
[fill] (25.5,-51) circle[radius=0.07] node(u3a9a2);
[fill] (25.5,-54) circle[radius=0.07] node(u3a9a3);
[fill] (28.5,-45) circle[radius=0.07] node(u3a10a1);
[fill] (28.5,-48) circle[radius=0.07] node(u3a10a2);
[fill] (28.5,-51) circle[radius=0.07] node(u3a10a3);
[fill] (28.5,-54) circle[radius=0.07] node(u3a10a4);
[fill] (31.5,-45) circle[radius=0.07] node(u3a11a1);
[fill] (31.5,-48) circle[radius=0.07] node(u3a11a2);
[fill] (31.5,-51) circle[radius=0.07] node(u3a11a3);
[fill] (31.5,-54) circle[radius=0.07] node(u3a11a4);
[fill] (34.5,-45) circle[radius=0.07] node(u3a12a1);
[fill] (34.5,-48) circle[radius=0.07] node(u3a12a2);
[fill] (34.5,-51) circle[radius=0.07] node(u3a12a3);
[fill] (34.5,-54) circle[radius=0.07] node(u3a12a4);

[->, green] (u3) to[out=-120,in=70] (u3a12a1);
[->,dotted,magenta] (u3a3a1)–(u3a2a1);
[->,red] (u3a2a1)–(u3a3a2);
[->,dotted,magenta] (u3a4a1)–(u3a3a1);
[->,dotted,magenta] (u3a4a2)–(u3a3a2);
[->,red] (u3a3a1)–(u3a4a2);
[->,dotted,magenta] (u3a5a1)–(u3a4a1);
[->,dotted,magenta] (u3a5a2)–(u3a4a2);
[->,red] (u3a4a1)–(u3a5a2);
[->,dotted,magenta] (u3a6a1)–(u3a5a1);
[->,dotted,magenta] (u3a6a2)–(u3a5a2);
[->,red] (u3a5a1)–(u3a6a2);
[->,dotted,magenta] (u3a7a1)–(u3a6a1);
[->,dotted,magenta] (u3a7a2)–(u3a6a2);
[->,red] (u3a6a1)–(u3a7a2);
[->,red] (u3a6a2)–(u3a7a3);
[->,dotted,magenta] (u3a8a1)–(u3a7a1);
[->,dotted,magenta] (u3a8a2)–(u3a7a2);
[->,dotted,magenta] (u3a8a3)–(u3a7a3);
[->,red] (u3a7a1)–(u3a8a2);
[->,red] (u3a7a2)–(u3a8a3);
[->,dotted,magenta] (u3a9a1)–(u3a8a1);
[->,dotted,magenta] (u3a9a2)–(u3a8a2);
[->,dotted,magenta] (u3a9a3)–(u3a8a3);
[->,red] (u3a8a1)–(u3a9a2);
[->,red] (u3a8a2)–(u3a9a3);
[->,dotted,magenta] (u3a10a1)–(u3a9a1);
[->,dotted,magenta] (u3a10a2)–(u3a9a2);
[->,dotted,magenta] (u3a10a3)–(u3a9a3);
[->,red] (u3a9a1)–(u3a10a2);
[->,red] (u3a9a2)–(u3a10a3);
[->,red] (u3a9a3)–(u3a10a4);
[->, blue] (u3a11a4)–(u3a10a4);
[->, blue] (u3a11a3)–(u3a10a3);
[->, blue] (u3a11a2)–(u3a10a2);
[->, blue] (u3a11a1)–(u3a10a1);
[->, blue] (u3a12a4)–(u3a11a4);
[->, blue] (u3a12a3)–(u3a11a3);
[->, blue] (u3a12a2)–(u3a11a2);
[->, blue] (u3a12a1)–(u3a11a1);
[->] (u3a10a1)–(u3a10a2);
[->] (u3a10a2)–(u3a10a3);
[->] (u3a10a3)–(u3a10a4);
[->] (u3a11a1)–(u3a11a2);
[->] (u3a11a2)–(u3a11a3);
[->] (u3a11a3)–(u3a11a4);
[->] (u3a12a1)–(u3a12a2);
[->] (u3a12a2)–(u3a12a3);
[->] (u3a12a3)–(u3a12a4);

[gray] (u1a10a4) ++(0.2, 0.2)–++(-3.4, 0)–++(-6, -6)–++(-3, 0)–++(-6, -6)–++(0, -0.4)–++(6.4, 0)–++(6, 6)–++(3, 0)–++(3, 3)–++(0, 3.4);

[gray] (u3a10a4) ++(0, -3) ++(0.2, 0.2)–++(-3.4, 0)–++(-6, -6)–++(-3, 0)–++(-6, -6)–++(0, -0.4)–++(6.4, 0)–++(6, 6)–++(3, 0)–++(3, 3)
–++(0, 3.4);

[gray] (u3a10a3) ++(-0.2, 0.2)–++(-9, -9)–++(-6, 0)–++(-9, -9)–++(0, -0.4)–++(3.4, 0)–++(9, 9)–++(3, 0)–++(6, 6)–++(3, 0)–++(0, 3.4)
–++(-0.4, 0);

[gray] (u2a10a2) ++(0, -3) ++(-0.2, 0.2)–++(-9, -9)–++(-6, 0)–++(-9, -9)–++(0, -0.4)–++(3.4, 0)–++(9, 9)–++(3, 0)–++(6, 6)–++(3, 0)
–++(0, 3.4)–++(-0.4, 0);

[gray, densely dotted, thick] (u3a10a3) ++(0.2, 0.2)–++(6, 0)–++(0, -3.4)–++(-6, 0);

[gray, densely dotted, thick] (u2a10a2) ++(0, -3) ++(0.2, 0.2)–++(6, 0)–++(0, -3.4)–++(-6, 0);

The butterfly diagram of a fixed point of a separated bow variety. The cyan dashed line indicates the boundary between the “left side" and “right side". Two different botched butterfly surgeries are outlined in solid gray. The surgery with site in the $U_1$ butterfly is subject to and satisfies constraint (<ref>). The other surgery is not subject to this constraint. The additional edges and vertices added to $\site$ to form $\overline\site$ are outlined in dotted gray.

Fix $p\in \bowvar(\brane)^\torus$ and consider its butterfly diagram. Note that the portion of the butterfly diagram below $X_n$ and the right side will be the same for all fixed points. Consider a surgery operation $\surgery$ as in Section <ref> on the left side of the butterfly diagram. That is, we take a $B, C, D$-invariant subgraph $\site$ of the left side of the $U_j$ butterfly and vertically translate it to stack it below the left side of the $U_{j'}$ butterfly, for some $1\leq j\neq j' \leq m$. Assume that the left side of the resulting diagram is the left side of some butterfly diagram (not necessarily associated to the same bow variety). Note that all butterfly surgeries meet these conditions. The key difference is that butterfly surgeries cannot alter the $X_n$ column. If $\site$ includes at least one vertex in the $X_n$ column, then call $\surgery$ a “botched butterfly surgery". See Figure <ref> for an example. In Section <ref>, the $B, C, D$-invariance of $\site$ emerged from other constraints on $\site$. Here, we must impose this property as an additional condition. We will produce a noncompact invariant curve from a botched butterfly surgery following the approach of Section <ref>.

The right side of the butterfly diagram is organized in a grid formed by horizontal leftward edges, which represent $A$ maps, and vertical downward edges, which represent $-B$ maps (see Figure <ref>). Let $\overline\site$ be the minimal $A, A^{-1}, B, C, D$-invariant subgraph containing $\site$. Let $\overline\site'$ be a copy of $\overline\site$ translated according to $\surgery$. Take the edges extending from the $U_{j'}$ butterfly to $\overline\site'$ imputed by the rules of Section <ref>, and create the corresponding edges between the $U_{j'}$ butterfly and $\overline\site$. As in Section <ref>, the resulting graph determines an element of $\MM$.

If $j>j'$, it is easy to verify the 0-momentum, S1, S2, and $\nu$ conditions, yielding an element $p_\gamma \in M$. If $j<j'$, all conditions will be satisfied except for the 0-momentum condition $B_{U_{j}}A_{U_{j}} - A_{U_{j}}B'_{U_{j}} + a_{U_{j}}b_{U_{j}} = 0$. Due to the structure of separated brane diagrams, the number of vertices in the $X_n, X_{n+1}, ..., X_{n+k-1}$ column of the $U_k$ butterfly is given by $c_k$. Recall that $c$ is the margin vector of D5 brane charges (see Section <ref>). Suppose $\site$ contains the bottom $r$ vertices of the $X_n$ column, where

\begin{equation} \label{eqn:typeII_constraint}
r \geq c_j - c_{j'} + 1.
\end{equation}

Label the vertices of the $X_{n+j-1}$ column of the $U_j$ butterfly from top to bottom as $v_1, ..., v_{c_j}$. Label the vertices of the $X_{n+j}$ column of the $U{j'}$ butterfly from top to bottom as $v'_1, ..., v'_{c_{j'}}$. Add additional edges according to the following rule:

* Create negative blue edges from $v'_{k + c_{j'} - c_j + r}$ to $v_k$ for $1 \leq k \leq c_j - r$. Recall that the blue edges represent $A$.

* Create a negative green edge from $v'_{c_{j'} - c_j + r}$ to the $U_j$ framing vertex. Recall that the green edge directed into the framing vertex represents $-b$ (Section <ref>).

This corrects the momentum, yielding an element $p_\gamma\in \bowvar(\brane)$. This rule is depicted in Figure <ref>. The point $p_\gamma$ is illustrated in Figure <ref> for both of the surgeries in Figure <ref>.

() at (0,-6) ;

(30,-3) circle[radius=0.14] node(u1);
[fill] (1.5,-33) circle[radius=0.07] node(u1a1a1);
[fill] (4.5,-30) circle[radius=0.07] node(u1a2a1);

[fill] (7.5,-27) circle[radius=0.07] node(u1a3a1);

[fill] (10.5,-24) circle[radius=0.07] node(u1a4a1);
[fill] (10.5,-27) circle[radius=0.07] node(u1a4a2);
[fill] (13.5,-21) circle[radius=0.07] node(u1a5a1);
[fill] (13.5,-24) circle[radius=0.07] node(u1a5a2);
[fill] (13.5,-27) circle[radius=0.07] node(u1a5a3);
[fill] (16.5,-18) circle[radius=0.07] node(u1a6a1);
[fill] (16.5,-21) circle[radius=0.07] node(u1a6a2);
[fill] (16.5,-24) circle[radius=0.07] node(u1a6a3);
[fill] (16.5,-27) circle[radius=0.07] node(u1a6a4);
[fill] (19.5,-15) circle[radius=0.07] node(u1a7a1);
[fill] (19.5,-18) circle[radius=0.07] node(u1a7a2);
[fill] (19.5,-21) circle[radius=0.07] node(u1a7a3);
[fill] (19.5,-24) circle[radius=0.07] node(u1a7a4);
[fill] (22.5,-12) circle[radius=0.07] node(u1a8a1);
[fill] (22.5,-15) circle[radius=0.07] node(u1a8a2);
[fill] (22.5,-18) circle[radius=0.07] node(u1a8a3);
[fill] (22.5,-21) circle[radius=0.07] node(u1a8a4);
[fill] (25.5,-9) circle[radius=0.07] node(u1a9a1);
[fill] (25.5,-12) circle[radius=0.07] node(u1a9a2);
[fill] (25.5,-15) circle[radius=0.07] node(u1a9a3);
[fill] (25.5,-18) circle[radius=0.07] node(u1a9a4);
[fill] (25.5,-21) circle[radius=0.07] node(u1a9a5);
[fill] (28.5,-6) circle[radius=0.07] node(u1a10a1);
[fill] (28.5,-9) circle[radius=0.07] node(u1a10a2);
[fill] (28.5,-12) circle[radius=0.07] node(u1a10a3);
[fill] (28.5,-15) circle[radius=0.07] node(u1a10a4);
[fill] (28.5,-18) circle[radius=0.07] node(u1a10a5);

[->, green] (u1) to[out=-120,in=70] (u1a10a1);
[->,dotted,magenta] (u1a2a1)–(u1a1a1);

[->,dotted,magenta] (u1a3a1)–(u1a2a1);

[->,dotted,magenta] (u1a4a1)–(u1a3a1);

[->,red] (u1a3a1)–(u1a4a2);
[->,dotted,magenta] (u1a5a1)–(u1a4a1);
[->,dotted,magenta] (u1a5a2)–(u1a4a2);
[->,red] (u1a4a1)–(u1a5a2);
[->,red] (u1a4a2)–(u1a5a3);
[->,dotted,magenta] (u1a6a1)–(u1a5a1);
[->,dotted,magenta] (u1a6a2)–(u1a5a2);
[->,dotted,magenta] (u1a6a3)–(u1a5a3);
[->,red] (u1a5a1)–(u1a6a2);
[->,red] (u1a5a2)–(u1a6a3);
[->,red] (u1a5a3)–(u1a6a4);
[->,dotted,magenta] (u1a7a1)–(u1a6a1);
[->,dotted,magenta] (u1a7a2)–(u1a6a2);
[->,dotted,magenta] (u1a7a3)–(u1a6a3);
[->,dotted,magenta] (u1a7a4)–(u1a6a4);
[->,red] (u1a6a1)–(u1a7a2);
[->,red] (u1a6a2)–(u1a7a3);
[->,red] (u1a6a3)–(u1a7a4);
[->,dotted,magenta] (u1a8a1)–(u1a7a1);
[->,dotted,magenta] (u1a8a2)–(u1a7a2);
[->,dotted,magenta] (u1a8a3)–(u1a7a3);
[->,dotted,magenta] (u1a8a4)–(u1a7a4);
[->,red] (u1a7a1)–(u1a8a2);
[->,red] (u1a7a2)–(u1a8a3);
[->,red] (u1a7a3)–(u1a8a4);
[->,dotted,magenta] (u1a9a1)–(u1a8a1);
[->,dotted,magenta] (u1a9a2)–(u1a8a2);
[->,dotted,magenta] (u1a9a3)–(u1a8a3);
[->,dotted,magenta] (u1a9a4)–(u1a8a4);
[->,red] (u1a8a1)–(u1a9a2);
[->,red] (u1a8a2)–(u1a9a3);
[->,red] (u1a8a3)–(u1a9a4);
[->,red] (u1a8a4)–(u1a9a5);
[->,dotted,magenta] (u1a10a1)–(u1a9a1);
[->,dotted,magenta] (u1a10a2)–(u1a9a2);
[->,dotted,magenta] (u1a10a3)–(u1a9a3);
[->,dotted,magenta] (u1a10a4)–(u1a9a4);
[->,dotted,magenta] (u1a10a5)–(u1a9a5);
[->,red] (u1a9a1)–(u1a10a2);
[->,red] (u1a9a2)–(u1a10a3);
[->,red] (u1a9a3)–(u1a10a4);
[->,red] (u1a9a4)–(u1a10a5);
[->] (u1a10a1)–(u1a10a2);
[->] (u1a10a2)–(u1a10a3);
[->] (u1a10a3)–(u1a10a4);
[->] (u1a10a4)–(u1a10a5);

() at (0,-45) ;

(36,-42) circle[radius=0.14] node(u3);
[fill] (4.5,-69) circle[radius=0.07] node(u3a2a1);
[fill] (7.5,-66) circle[radius=0.07] node(u3a3a1);
[fill] (7.5,-69) circle[radius=0.07] node(u3a3a2);
[fill] (10.5,-63) circle[radius=0.07] node(u3a4a1);
[fill] (10.5,-66) circle[radius=0.07] node(u3a4a2);
[fill] (13.5,-60) circle[radius=0.07] node(u3a5a1);
[fill] (13.5,-63) circle[radius=0.07] node(u3a5a2);
[fill] (16.5,-57) circle[radius=0.07] node(u3a6a1);
[fill] (16.5,-60) circle[radius=0.07] node(u3a6a2);
[fill] (19.5,-54) circle[radius=0.07] node(u3a7a1);
[fill] (19.5,-57) circle[radius=0.07] node(u3a7a2);
[fill] (19.5,-60) circle[radius=0.07] node(u3a7a3);
[fill] (22.5,-51) circle[radius=0.07] node(u3a8a1);
[fill] (22.5,-54) circle[radius=0.07] node(u3a8a2);
[fill] (22.5,-57) circle[radius=0.07] node(u3a8a3);
[fill] (25.5,-48) circle[radius=0.07] node(u3a9a1);
[fill] (25.5,-51) circle[radius=0.07] node(u3a9a2);
[fill] (25.5,-54) circle[radius=0.07] node(u3a9a3);
[fill] (28.5,-45) circle[radius=0.07] node(u3a10a1);
[fill] (28.5,-48) circle[radius=0.07] node(u3a10a2);
[fill] (28.5,-51) circle[radius=0.07] node(u3a10a3);
[fill] (28.5,-54) circle[radius=0.07] node(u3a10a4);
[fill] (31.5,-45) circle[radius=0.07] node(u3a11a1);
[fill] (31.5,-48) circle[radius=0.07] node(u3a11a2);
[fill] (31.5,-51) circle[radius=0.07] node(u3a11a3);
[fill] (31.5,-54) circle[radius=0.07] node(u3a11a4);
[fill] (34.5,-45) circle[radius=0.07] node(u3a12a1);
[fill] (34.5,-48) circle[radius=0.07] node(u3a12a2);
[fill] (34.5,-51) circle[radius=0.07] node(u3a12a3);
[fill] (34.5,-54) circle[radius=0.07] node(u3a12a4);

[->, green] (u3) to[out=-120,in=70] (u3a12a1);
[->,dotted,magenta] (u3a3a1)–(u3a2a1);
[->,red] (u3a2a1)–(u3a3a2);
[->,dotted,magenta] (u3a4a1)–(u3a3a1);
[->,dotted,magenta] (u3a4a2)–(u3a3a2);
[->,red] (u3a3a1)–(u3a4a2);
[->,dotted,magenta] (u3a5a1)–(u3a4a1);
[->,dotted,magenta] (u3a5a2)–(u3a4a2);
[->,red] (u3a4a1)–(u3a5a2);
[->,dotted,magenta] (u3a6a1)–(u3a5a1);
[->,dotted,magenta] (u3a6a2)–(u3a5a2);
[->,red] (u3a5a1)–(u3a6a2);
[->,dotted,magenta] (u3a7a1)–(u3a6a1);
[->,dotted,magenta] (u3a7a2)–(u3a6a2);
[->,red] (u3a6a1)–(u3a7a2);
[->,red] (u3a6a2)–(u3a7a3);
[->,dotted,magenta] (u3a8a1)–(u3a7a1);
[->,dotted,magenta] (u3a8a2)–(u3a7a2);
[->,dotted,magenta] (u3a8a3)–(u3a7a3);
[->,red] (u3a7a1)–(u3a8a2);
[->,red] (u3a7a2)–(u3a8a3);
[->,dotted,magenta] (u3a9a1)–(u3a8a1);
[->,dotted,magenta] (u3a9a2)–(u3a8a2);
[->,dotted,magenta] (u3a9a3)–(u3a8a3);
[->,red] (u3a8a1)–(u3a9a2);
[->,red] (u3a8a2)–(u3a9a3);
[->,dotted,magenta] (u3a10a1)–(u3a9a1);
[->,dotted,magenta] (u3a10a2)–(u3a9a2);
[->,dotted,magenta] (u3a10a3)–(u3a9a3);
[->,red] (u3a9a1)–(u3a10a2);
[->,red] (u3a9a2)–(u3a10a3);
[->,red] (u3a9a3)–(u3a10a4);
[->, blue] (u3a11a4)–(u3a10a4);
[->, blue] (u3a11a3)–(u3a10a3);
[->, blue] (u3a11a2)–(u3a10a2);
[->, blue] (u3a11a1)–(u3a10a1);
[->, blue] (u3a12a4)–(u3a11a4);
[->, blue] (u3a12a3)–(u3a11a3);
[->, blue] (u3a12a2)–(u3a11a2);
[->, blue] (u3a12a1)–(u3a11a1);
[->] (u3a10a1)–(u3a10a2);
[->] (u3a10a2)–(u3a10a3);
[->] (u3a10a3)–(u3a10a4);
[->] (u3a11a1)–(u3a11a2);
[->] (u3a11a2)–(u3a11a3);
[->] (u3a11a3)–(u3a11a4);
[->] (u3a12a1)–(u3a12a2);
[->] (u3a12a2)–(u3a12a3);
[->] (u3a12a3)–(u3a12a4);

[->] (u3a10a4)to[out=85, in=-85](u1a10a4);
[magenta, dotted, ->] (u3a10a4)to[out=225, in=45](u1a9a3); [magenta, dotted, ->] (u3a7a3)to[out=225, in=45](u1a6a2);
[red, ->] (u3a8a3)to[out=0, in=180](u1a9a3); [red, ->] (u3a7a3)to[out=0, in=180](u1a8a3);
[red, ->] (u3a5a2)to[out=0, in=180](u1a6a2); [red, ->] (u3a4a2)to[out=0, in=180](u1a5a2); [red, ->] (u3a3a2)to[out=0, in=180](u1a4a2);
[blue, ->] (u3a11a4)–(u1a10a3); [blue, ->] (u3a11a3)–(u1a10a2); [blue, ->] (u3a11a2)–(u1a10a1) node[pos=0.75, right]$-1$;
[green, ->] (u3a11a1)–(u1) node[pos=0.75, right]$-1$;

() at (0,-33) ;

(33,-30) circle[radius=0.14] node(u2);
[fill] (16.5,-45) circle[radius=0.07] node(u2a6a1);
[fill] (19.5,-42) circle[radius=0.07] node(u2a7a1);
[fill] (19.5,-45) circle[radius=0.07] node(u2a7a2);
[fill] (22.5,-39) circle[radius=0.07] node(u2a8a1);
[fill] (22.5,-42) circle[radius=0.07] node(u2a8a2);
[fill] (25.5,-36) circle[radius=0.07] node(u2a9a1);
[fill] (25.5,-39) circle[radius=0.07] node(u2a9a2);
[fill] (28.5,-33) circle[radius=0.07] node(u2a10a1);
[fill] (28.5,-36) circle[radius=0.07] node(u2a10a2);
[fill] (31.5,-33) circle[radius=0.07] node(u2a11a1);
[fill] (31.5,-36) circle[radius=0.07] node(u2a11a2);

[->, green] (u2) to[out=-120,in=70] (u2a11a1);
[->,dotted,magenta] (u2a7a1)–(u2a6a1);
[->,red] (u2a6a1)–(u2a7a2);
[->,dotted,magenta] (u2a8a1)–(u2a7a1);
[->,dotted,magenta] (u2a8a2)–(u2a7a2);
[->,red] (u2a7a1)–(u2a8a2);
[->,dotted,magenta] (u2a9a1)–(u2a8a1);
[->,dotted,magenta] (u2a9a2)–(u2a8a2);
[->,red] (u2a8a1)–(u2a9a2);
[->,dotted,magenta] (u2a10a1)–(u2a9a1);
[->,dotted,magenta] (u2a10a2)–(u2a9a2);
[->,red] (u2a9a1)–(u2a10a2);
[->, blue] (u2a11a2)–(u2a10a2);
[->, blue] (u2a11a1)–(u2a10a1);
[->] (u2a10a1)–(u2a10a2);
[->] (u2a11a1)–(u2a11a2);

() at (0,-45) ;

(36,-42) circle[radius=0.14] node(u3);
[fill] (4.5,-69) circle[radius=0.07] node(u3a2a1);
[fill] (7.5,-66) circle[radius=0.07] node(u3a3a1);
[fill] (7.5,-69) circle[radius=0.07] node(u3a3a2);
[fill] (10.5,-63) circle[radius=0.07] node(u3a4a1);
[fill] (10.5,-66) circle[radius=0.07] node(u3a4a2);
[fill] (13.5,-60) circle[radius=0.07] node(u3a5a1);
[fill] (13.5,-63) circle[radius=0.07] node(u3a5a2);
[fill] (16.5,-57) circle[radius=0.07] node(u3a6a1);
[fill] (16.5,-60) circle[radius=0.07] node(u3a6a2);
[fill] (19.5,-54) circle[radius=0.07] node(u3a7a1);
[fill] (19.5,-57) circle[radius=0.07] node(u3a7a2);
[fill] (19.5,-60) circle[radius=0.07] node(u3a7a3);
[fill] (22.5,-51) circle[radius=0.07] node(u3a8a1);
[fill] (22.5,-54) circle[radius=0.07] node(u3a8a2);
[fill] (22.5,-57) circle[radius=0.07] node(u3a8a3);
[fill] (25.5,-48) circle[radius=0.07] node(u3a9a1);
[fill] (25.5,-51) circle[radius=0.07] node(u3a9a2);
[fill] (25.5,-54) circle[radius=0.07] node(u3a9a3);
[fill] (28.5,-45) circle[radius=0.07] node(u3a10a1);
[fill] (28.5,-48) circle[radius=0.07] node(u3a10a2);
[fill] (28.5,-51) circle[radius=0.07] node(u3a10a3);
[fill] (28.5,-54) circle[radius=0.07] node(u3a10a4);
[fill] (31.5,-45) circle[radius=0.07] node(u3a11a1);
[fill] (31.5,-48) circle[radius=0.07] node(u3a11a2);
[fill] (31.5,-51) circle[radius=0.07] node(u3a11a3);
[fill] (31.5,-54) circle[radius=0.07] node(u3a11a4);
[fill] (34.5,-45) circle[radius=0.07] node(u3a12a1);
[fill] (34.5,-48) circle[radius=0.07] node(u3a12a2);
[fill] (34.5,-51) circle[radius=0.07] node(u3a12a3);
[fill] (34.5,-54) circle[radius=0.07] node(u3a12a4);

[->, green] (u3) to[out=-120,in=70] (u3a12a1);
[->,dotted,magenta] (u3a3a1)–(u3a2a1);
[->,red] (u3a2a1)–(u3a3a2);
[->,dotted,magenta] (u3a4a1)–(u3a3a1);
[->,dotted,magenta] (u3a4a2)–(u3a3a2);
[->,red] (u3a3a1)–(u3a4a2);
[->,dotted,magenta] (u3a5a1)–(u3a4a1);
[->,dotted,magenta] (u3a5a2)–(u3a4a2);
[->,red] (u3a4a1)–(u3a5a2);
[->,dotted,magenta] (u3a6a1)–(u3a5a1);
[->,dotted,magenta] (u3a6a2)–(u3a5a2);
[->,red] (u3a5a1)–(u3a6a2);
[->,dotted,magenta] (u3a7a1)–(u3a6a1);
[->,dotted,magenta] (u3a7a2)–(u3a6a2);
[->,red] (u3a6a1)–(u3a7a2);
[->,red] (u3a6a2)–(u3a7a3);
[->,dotted,magenta] (u3a8a1)–(u3a7a1);
[->,dotted,magenta] (u3a8a2)–(u3a7a2);
[->,dotted,magenta] (u3a8a3)–(u3a7a3);
[->,red] (u3a7a1)–(u3a8a2);
[->,red] (u3a7a2)–(u3a8a3);
[->,dotted,magenta] (u3a9a1)–(u3a8a1);
[->,dotted,magenta] (u3a9a2)–(u3a8a2);
[->,dotted,magenta] (u3a9a3)–(u3a8a3);
[->,red] (u3a8a1)–(u3a9a2);
[->,red] (u3a8a2)–(u3a9a3);
[->,dotted,magenta] (u3a10a1)–(u3a9a1);
[->,dotted,magenta] (u3a10a2)–(u3a9a2);
[->,dotted,magenta] (u3a10a3)–(u3a9a3);
[->,red] (u3a9a1)–(u3a10a2);
[->,red] (u3a9a2)–(u3a10a3);
[->,red] (u3a9a3)–(u3a10a4);
[->, blue] (u3a11a4)–(u3a10a4);
[->, blue] (u3a11a3)–(u3a10a3);
[->, blue] (u3a11a2)–(u3a10a2);
[->, blue] (u3a11a1)–(u3a10a1);
[->, blue] (u3a12a4)–(u3a11a4);
[->, blue] (u3a12a3)–(u3a11a3);
[->, blue] (u3a12a2)–(u3a11a2);
[->, blue] (u3a12a1)–(u3a11a1);
[->] (u3a10a1)–(u3a10a2);
[->] (u3a10a2)–(u3a10a3);
[->] (u3a10a3)–(u3a10a4);
[->] (u3a11a1)–(u3a11a2);
[->] (u3a11a2)–(u3a11a3);
[->] (u3a11a3)–(u3a11a4);
[->] (u3a12a1)–(u3a12a2);
[->] (u3a12a2)–(u3a12a3);
[->] (u3a12a3)–(u3a12a4);

[magenta, dotted, ->] (u2a6a1)to[out=225, in=45](u3a5a1); [magenta, dotted, ->] (u2a7a2)to[out=225, in=45](u3a6a2);
[red, ->] (u2a7a2)to[out=0, in=180](u3a8a3); [red, ->] (u2a8a2)to[out=0, in=180](u3a9a3); [red, ->] (u2a9a2)to[out=0, in=180](u3a10a3);
[->] (u2a10a2)to[out=-75, in=75](u3a10a3); [->] (u2a11a2)to[out=-75, in=75](u3a11a3);

Construction of the invariant curves corresponding to the two botched butterfly surgeries in Figure <ref>.

Using similar arguments to those in Section <ref>, one can show that the orbit of $p_\gamma$ is 1-dimensional, and its closure $\gamma$ is an invariant curve containing $p$. Additionally, if $\site$ has $k$ connected components, then a $k$-dimensional pencil $\pen_\surgery$ of invariant curves containing $p$ can be constructed. These results are summarized by the following analog to Lemma <ref>. The question of compactness is postponed until Section <ref>.

Fix a separated bow variety $\bowvar(\brane)$ and $p\in \bowvar(\brane)^\torus$. Let $\surgery$ be a botched butterfly surgery moving the site $\site$ from the $U_j$ butterfly to the $U_{j'}$ butterfly. If $j < j'$, impose the additional condition that $\site$ contains at least $c_j - c_{j'} + 1$ vertices of the $X_n$ column, where $c$ is the margin vector of D5 brane charges. Then, there exists a $k$-dimensional pencil $\pen_\surgery$ of $\torus$-invariant curves containing $p$. Each curve has tangent weight

\[
\frac{\ub}{\ub'}\hb^{-\Delta_\surgery y},
\]

at $p$.

§.§ Nonsurgery Curves

We retain the conventions of Section <ref> for this section. Fix $1\leq j < j' \leq m$. We will construct $\max\{0, c_{j'} - c_j\}$ many invariant curves. To construct the $i$th curve,

* create a green edge from $v'_i$ to the $U_j$ framing vertex,

* create blue edges from $v'_{i+k}$ to $v_k$ for $1 \leq k \leq c_1$.

Figure <ref> depicts these added edges. Using the arguments of Section <ref>, one can show that this results in a point of $\bowvar(\brane)$ with a 1-dimensional orbit having $p$ in its closure. In this case, the rank of $b$ increases with the added edges, ruling out the orbit being equal to $p$. The tangent weight of this curve at $p$ can easily be determined by examining the added $b$ edge and comparing the heights of the target and source.

botched surgery [0.4]

(0, 0) node(l) ++(2, 0) node(u) ++(2, 0) node(r);
[blue] (u) +(0.5, -0.5) – +(-0.5, 0.5) +(0, 0.5) node$U_j$;

(2, -1) circle[radius = .05] node(uj);

(1, -1.5) node(a1) ++(0, -.5) node(a2) ++(0, -.5) node(a3) ++(0, -.5) node(a4) ++(0, -.5) node(a5) ++(0, -.5) node(a6)
++(0, -.5) node(a7) ++(0, -.5) node(a8);

(a8) ++(0, -1) node(b1) ++(0, -.5) node(b2) ++(0, -.5) node(b3) ++(0, -.5) node(b4) ++(0, -.5) node(b5) ++(0, -.5) node(b6)
++(0, -.5) node(b7) ++(0, -.5) node(b8);

(b1) ++(2, 0) node(c1) ++(0, -.5) node(c2) ++(0, -.5) node(c3) ++(0, -.5) node(c4) ++(0, -.5) node(c5) ++(0, -.5) node(c6)
++(0, -.5) node(c7) ++(0, -.5) node(c8);

[fill] (a1) circle[radius = 0.025]; [fill] (a4) circle [radius = 0.025]; [fill] (a5) circle [radius = 0.025]; [fill] (a8) circle [radius = 0.025];
[fill] (b1) circle[radius = 0.025]; [fill] (b4) circle[radius = 0.025]; [fill] (b5) circle[radius = 0.025]; [fill] (b8) circle[radius = 0.025];
[fill] (c1) circle[radius = 0.025]; [fill] (c4) circle[radius = 0.025]; [fill] (c5) circle[radius = 0.025]; [fill] (c8) circle[radius = 0.025];

[->, green] (uj) to[out = 250, in = 45] (a1);
[->] (a1) – (a2); [->] (a3) – (a4); [->] (a4) – (a5); [->] (a5) – (a6); [->] (a7) – (a8);
[->] (b1) – (b2); [->] (b3) – (b4); [->] (b4) – (b5); [->] (b5) – (b6); [->] (b7) – (b8);
[->] (c1) – (c2); [->] (c3) – (c4); [->] (c4) – (c5); [->] (c5) – (c6); [->] (c7) – (c8);

[->, blue] (c1) – (b1); [->, blue] (c4) – (b4); [->, blue] (c5) – (b5); [->, blue] (c8) – (b8);

[dotted] (a2) – (a3); [dotted] (a6) – (a7);
[dotted] (b2) – (b3); [dotted] (b6) – (b7);
[dotted] (c2) – (c3); [dotted] (c6) – (c7);

[->, green] (c4) to[out = 75, in = 280] (uj); (2.6, -2.2) node[green]-1;
[->, blue] (c5) – (a1) node[pos = 0.75, right]-1; [->, blue] (c8) – (a4) node[pos = 0.75, right]-1;
[->] (b8) to[out = 105, in = 255] (a5);

[gray] ($(a5)+(-0.1, 0.1)$) – ($(a8)+(-0.1, -0.1)$) – ($(a8)+(0.1, -0.1)$) – ($(a5)+(0.1, 0.1)$) – ($(a5)+(-0.1, 0.1)$);

nonsurgery [0.4]

(0, 0) node(l) ++(2, 0) node(u) ++(2, 0) node(r);
[blue] (u) +(0.5, -0.5) – +(-0.5, 0.5) +(0, 0.5) node$U_j$;

(2, -1) circle[radius = .05] node(uj);

(1, -1.5) node(a1) ++(0, -.5) node(a2) ++(0, -.5) node(a3) ++(0, -.5) node(a4) ++(0, -.5);

(a4) ++(0, -1.5) node(b1) ++(0, -.5) node(b2) ++(0, -.5) node(b3) ++(0, -.5) node(b4) ++(0, -.5) node(b5) ++(0, -.5) node(b6)
++(0, -.5) node(b7) ++(0, -.5) node(b8) ++(0, -.5) node(b9) ++(0, -.5) node(b10) ++(0, -.5) node(b11);

(b1) ++(2, 0) node(c1) ++(0, -.5) node(c2) ++(0, -.5) node(c3) ++(0, -.5) node(c4) ++(0, -.5) node(c5) ++(0, -.5) node(c6)
++(0, -.5) node(c7) ++(0, -.5) node(c8) ++(0, -.5) node(c9) ++(0, -.5) node(c10) ++(0, -.5) node(c11);

[fill] (a1) circle[radius = 0.025]; [fill] (a4) circle [radius = 0.025];

[fill] (b1) circle[radius = 0.025]; [fill] (b4) circle[radius = 0.025]; [fill] (b5) circle[radius = 0.025]; [fill] (b8) circle[radius = 0.025];
[fill] (b11) circle [radius = 0.025];

[fill] (c1) circle[radius = 0.025]; [fill] (c4) circle[radius = 0.025]; [fill] (c5) circle[radius = 0.025]; [fill] (c8) circle[radius = 0.025];
[fill] (c11) circle [radius = 0.025];

[->, green] (uj) to[out = 250, in = 45] (a1);

[->] (a1) – (a2); [dotted] (a2) – (a3); [->] (a3) – (a4);

[->] (b1) – (b2); [dotted] (b2) – (b3); [->] (b3) – (b4); [->] (b4) – (b5); [->] (b5) – (b6); [dotted] (b6) – (b7);
[->] (b7) – (b8); [->] (b8) – (b9); [dotted] (b9) – (b10); [->] (b10) – (b11);

[->] (c1) – (c2); [dotted] (c2) – (c3); [->] (c3) – (c4); [->] (c4) – (c5); [->] (c4) – (c5); [->] (c5) – (c6);
[dotted] (c6) – (c7); [->] (c7) – (c8); [->] (c8) – (c9); [dotted] (c9) – (c10); [->] (c10) – (c11);

[->, blue] (c1) – (b1); [->, blue] (c4) – (b4); [->, blue] (c5) – (b5); [->, blue] (c8) – (b8);  [->, blue] (c11) – (b11);

[->, green] (c4) to[out = 75, in = 280] (uj);
[->, blue] (c5) – (a1); [->, blue] (c8) – (a4);

Local depictions of (a) the curve associated to a botched surgery moving a part of a higher butterfly to a lower butterfly and (b) a nonsurgery invariant curve. In (a), those visible vertices that are contained in $\overline\site$ are outlined in gray.

Let $\bowvar(\brane)$ be a separated bow variety and let $c = (c_1, ..., c_m)$ denote the margin vector of D5 brane charges. Let $p\in \bowvar(\brane)^\torus$, and $1\leq j < j' \leq m$. Then, there are $\max\{0, c_{j'} - c_j\}$ invariant curves containing $p$ with tangent weights

\[
\frac{\ub_j}{\ub_{j'}}\hb, \frac{\ub_j}{\ub_{j'}}\hb^2, ..., \frac{\ub_j}{\ub_{j'}}\hb^{\max\{0, c_{j'} - c_j\}}
\]

at $p$.

§.§ Young Diagram Surgery

The combinatorics of Section <ref> and Section <ref> can be reformulated using the more familiar language of partitions and Young diagrams. Again, we fix a separated bow variety $\bowvar(\brane)$ and a fixed point $p\in \bowvar(\brane)^\torus$. Let $M$ be the associated BCT and $c=(c_1, ..., c_m)$ be the margin vector of D5 brane charges. For each D5 brane $U_j$, define a partition $\lambda^{(j)}$ with $c_j$ distinct parts $\lambda^{(j)}_1 > \cdots > \lambda^{(j)}_{c_i} > 0$ by

* letting $i_1 < i_2 < \cdots < i_{c_j}$ be the indices for which $(i_k, j) = 1$ and

* setting $\lambda^{(j)}_k = n - i_k + 1$.

In other words, the parts of the partition correspond to the vertical positions of the 1's in the $j$th column of $M$ measured from the bottom. The Young diagrams associated with these partitions will be drawn in a nonstandard way with the boxes aligned on the right. Each row of boxes represents a part of the partition, and the longer rows will be drawn on top. See Figure <ref> for an example. We will use $\lambda^{(j)}$ to denote the partition as well as its Young diagram.

\begin{pmatrix}
\end{pmatrix}

[scale=0.35, xscale=-1]
(0, 0)–(10, 0); (0, -1)–(10, -1); (0, -2)–(7, -2); (0, -3)–(6, -3); (0, -4)–(5, -4); (0, -5)–(2, -5);
(0, 0)–(0, -5); (1, 0)–(1, -5); (2, 0)–(2, -5); (3, 0)–(3, -4); (4, 0)–(4, -4); (5, 0)–(5, -4); (6, 0)–(6, -3); (7, 0)–(7, -2); (8, 0)–(8, -1); (9, 0)–(9, -1); (10, 0)–(10, -1);

(0, -6)–(5, -6); (0, -7)–(5, -7); (0, -8)–(4, -8);