#  $\mathbb{C}$-boundary links up to six crossings

S. Yu. Orevkov  Steklov Mathematical Institute of Russian Academy of Sciences,
Moscow, Russia.  IMT, l’université Paul Sabatier, 118 route de Narbonne,
Toulouse, France.

An oriented link is called $\mathbb{C}$-boundary if it is realizable as
$(\partial B,A\cap\partial B)$ where $A$ is an algebraic curve in
$\mathbb{C}^{2}$ and $B$ is an embedded $4$-ball. This notion was introduced
by Michel Boileau and Lee Rudolph in 1995. In a recent joint paper with N. G.
Kruzhilin we gave a complete classification of $\mathbb{C}$-boundaries with at
most 5 crossings. In the present paper a more regular method of construction
of $\mathbb{C}$-boundaries is proposed and the classification is extended up
to 6 crossings.

## 1\. Introduction

Boileau and Rudolph [2] defined $\mathbb{C}$-boundary as an oriented link in
the $3$-sphere $\mathbb{S}^{3}$ which can be realized as $(\partial
B,A\cap\partial B)$ where $A$ is an algebraic curve in $\mathbb{C}^{2}$ and
$B$ is an embedded $4$-ball (with the boundary orientation of $A\cap\partial
B$ induced from $A\cap B$). It is observed in [2] that Kronheimer-Mrowka
Theorem [6] (former Thom Conjecture) implies that some knots are not
$\mathbb{C}$-boundaries, for example, the figure-eight knot $4_{1}$.

If $B$ is pseudo-convex (for example, a standard $4$-ball), then $(\partial
B,A\cap\partial B)$ is a quasipositive link by [1]. In [7] we observed that
there exist non-quasipositive $\mathbb{C}$-boundaries, the simplest ones being
$L\,\\#\,{-L^{*}}$ where $L$ is a quasipositive link. Here and below, $L^{*}$
denotes the mirror image of $L$, and $-L$ denotes $L$ with the opposite
orientation. In [7] we also corrected some errors in [2] and observed that it
is often more efficient to apply the so-called Immersed Thom Conjecture in
order to obtain restrictions on the $\mathbb{C}$-boundaries (see details in
[7, §3] and in §3 here).

It is natural to distinguish the case when $L$ is realizable as $A\cap\partial
B$ as above, and moreover $A\setminus B$ is connected. Such links are called
in [7] strong $\mathbb{C}$-boundaries. Examples of non-strong
$\mathbb{C}$-boundaries are given in [7]. The simplest one is the split sum
$2_{1}\sqcup 2_{1}^{*}$ (we denote the positive two-component Hopf link by
$2_{1}$).

We gave in [7] a complete list of $\mathbb{C}$-boundaries and strong
$\mathbb{C}$-boundaries with at most 5 crossings (including split and
composite links). In the present paper a complete list of
$\mathbb{C}$-boundaries with 6 crossings is given (see §3). For two of them it
remains unknown if they are strong $\mathbb{C}$-boundaries. Very elementary
constructions were used in [7]. They are not enough for links with 6
crossings, and in §2 we present a more systematic way to construct (strong)
$\mathbb{C}$-boundaries. In contrary, the fact that some links are not
(strong) $\mathbb{C}$-boundaries, is proven here by the same methods as in
[7]. In §4.2 we show that 10 (among 15) slice knots with $\leq 9$ crossings
are $\mathbb{C}$-boundaries. Some open questions are posed in §4.3.

Acknowledgement. I thank Michel Boileau and the anonymous referee for valuable
remarks.

## 2\. A construction of $\mathbb{C}$-boundaries

### 2.1. Notation and terminology

We denote the segment $[0,1]$ by $I$ and we set $I^{2}=I\times I$. Let $L$ be
an oriented link. A band attached to $L$ is an embedding
$b:I^{2}\to\mathbb{S}^{3}$ such that $b|_{I\times\partial I}\to L$ is an
orientation preserving embedding where the orientation of $I\times\partial I$
is inherited from the boundary orientation of $\partial(I^{2})$ (see Figure
1). Sometimes, abusing the language, we use this term for $b(I^{2})$ rather
than for $b$. The result of the band-move (or band surgery) on $L$ along $b$
is

$L_{b}=\big{(}L\setminus b(I\times\partial I)\big{)}\cup b(\partial I\times
I)$

with the orientation coherent with $L$ (see Figure 1). We say that two bands
$b,b^{\prime}$ are disjoint if $b(I^{2})\cap b^{\prime}(I^{2})=\varnothing$.
If $\mathbb{b}$ is a collection of pairwise disjoint bands attached to $L$,
let $L_{\mathbb{b}}$ denote the result of simultaneous band-moves on $L$ along
all these bands.

A band $b$ attached to a link $L$ is called trivial if $L_{b}$ is a split sum
$L^{\prime}\sqcup O$ where $O$ is an unknot composed of an arc of $\partial b$
and an arc of $L$ (see the right panel in Figure 1). In this case, $O\cap L$
is called the vanishing arc of $(L,\mathbb{b})$. A collection
$\mathbb{b}=\\{b_{1},\dots,b_{n}\\}$ of pairwise disjoint bands attached to
$L$ is trivial if all $b_{i}$ are trivial for $L$ and their vanishing arcs are
pairwise disjoint. When such $\mathbb{b}$ is a subset of a larger collection
$\mathbb{b}^{\prime}$ of pairwise disjoint bands, we say that $\mathbb{b}$ is
a trivial subset of $\mathbb{b}^{\prime}$, if it is a trivial collection of
bands and all its vanishing arcs are disjoint from the bands belonging to
$\mathbb{b}^{\prime}\setminus\mathbb{b}$.

Figure 1 A band $b$ attached to a link $L$. The link $L_{b}$.  In the middle:
positive $\theta$-monotone band. On the right: trivial band.

Now we identify $\mathbb{S}^{3}$ with a one-point compactification of
$\mathbb{R}^{3}$ with standard coordinates $(x,y,z)$. Let us fix cylindrical
coordinates $(r,\theta,z)$ in $\mathbb{R}^{3}$, that is $x=r\cos\theta$,
$y=r\sin\theta$. We say that a link $L$ is $\theta$-monotone if $r|_{L}>0$ and
$d\theta$ is positive on $L$. We also call such links geometric closed braids.

Let $L$ be a $\theta$-monotone link. A band $b$ attached to $L$ is called
positive $\theta$-monotone if $L_{b}$ is $\theta$-monotone and the projection
of some neighborhood of $L\cup b(I^{2})$ to the $(\theta,z)$-plane looks as in
the middle panel of Figure 1.

An isotopy $\\{L_{t}\\}_{t\in I}$ is called $\theta$-monotone if each link
$L_{t}$ is $\theta$-monotone.

### 2.2. The construction

Let $L$ be a $\theta$-monotone link which is the braid closure of a
quasipositive (maybe, trivial) braid, and let $\mathbb{b}$ be a collection of
pairwise disjoint bands attached to $L$. Suppose that there exists a sequence
of links $L=L_{0},L_{1},\dots,L_{n}=L^{\prime}$ and a sequence of collections
of bands
$\mathbb{b}=\mathbb{b}_{0},\mathbb{b}_{1},\dots,\mathbb{b}_{n}=\mathbb{b}^{\prime}$
($\mathbb{b}_{i}$ is attached to $L_{i}$) such that for each $i=1,\dots,n$,
one of the following two conditions holds:

######  Proposition \propConstr

If $\mathbb{b}^{\prime}$ is trivial, then $L_{\mathbb{b}}$ is a
$\mathbb{C}$-boundary.

###### Demonstration Proof

By construction, $L^{\prime}$ is the braid closure of a quasipositive braid.
Hence, by Rudolph’s theorem [13], $(\mathbb{S}^{3},L^{\prime})$ is
diffeomorphic to $(\partial B^{\prime},A\cap\partial B^{\prime})$,
$B^{\prime}=\Delta^{\prime}\times\Delta^{\prime\prime}$, where
$\Delta^{\prime}$ and $\Delta^{\prime\prime}$ are some disks in $\mathbb{C}$,
and $A$ is a complex algebraic curve in $\mathbb{C}^{2}$ disjoint from
$\Delta^{\prime}\times\partial\Delta^{\prime\prime}$. Moreover, the arguments
in [13] imply that there exist embedded disks
$\Delta=\Delta_{0}\subset\Delta_{1}\subset\dots\subset\Delta_{n}=\Delta^{\prime}$
(with $\Delta_{i-1}\subset\operatorname{Int}\Delta_{i}$) such that
$(B_{i},A\cap\partial B_{i})\approx(\mathbb{S}^{3},L_{i})$, $i=0,\dots,n$,
where $B_{i}=\Delta_{i}\times\Delta^{\prime\prime}$, and each time when
$L_{i}$ is obtained from $L_{i-1}$ by a positive $\theta$-monotone band-move,
the annulus $\Delta_{i}\setminus\Delta_{i-1}$ contains a ramification point of
$\pi|_{A\cap B^{\prime}}$ where
$\pi:B^{\prime}=\Delta^{\prime}\times\Delta^{\prime\prime}\to\Delta^{\prime}$
is the projection onto the first factor.

We include the disk boundaries $\partial\Delta_{0},\dots,\partial\Delta_{n}$
into a continuous family. Namely, consider a diffeomorphism
$f:\mathbb{S}^{1}\times I\to\Delta^{\prime}\setminus\operatorname{Int}\Delta$,
such that $\partial\Delta_{i}=f_{t_{i}}(\mathbb{S}^{1})$ for some numbers
$t_{i}$ in the range $0=t_{0}<t_{1}<\dots<t_{n}=1$ (here
$f_{t}:\mathbb{S}^{1}\to\Delta^{\prime}$ is defined by $f_{t}(p)=f(p,t)$).

For each band $b$ from $\mathbb{b}$, the isotopies between $\mathbb{b}_{i-1}$
and $\mathbb{b}_{i}$ (which are identical when
$\mathbb{b}_{i}=(\mathbb{b}_{i-1})_{c_{i}}$) can be glued together into a
single isotopy $\\{b_{t}:I^{2}\to B^{\prime}\\}_{t\in I}$ such that
$\pi(b_{t}(I^{2}))\subset f_{t}(\mathbb{S}^{1})$. Let $\\{h_{t}:I^{3}\to
B^{\prime}\\}_{t\in I}$ be an isotopy obtained from $\\{b_{t}\\}$ by replacing
each band $b_{t}:I^{2}\to\pi^{-1}(f_{t}(\mathbb{S}^{1}))$ with it thickening
$h_{t}:I^{3}\to\pi^{-1}(f_{t}(\mathbb{S}^{1}))$ as shown in Figure 2. Finally,
we modify this isotopy near $t=1$ as shown in Figure 3 (still keeping the
notation $h_{t}$ for it).

Figure 2 A thickened band.

Figure 3 The final isotopy of a thickened trivial band.

We obtain an embedding of the 4-cube $h=h^{(b)}:I^{4}=I^{3}\times I\to
B^{\prime}$, $h(p,t)=h_{t}(p)$. The intersection of $h(I^{4})$ with
$B=\Delta\times\Delta^{\prime\prime}$ is the image of one of the eight faces
of $I^{4}$, that is $h(I^{4})\cap B=h(I^{3}\times 0)$. Thus the following set
$\hat{B}$ (appropriately smoothed) is an embedded 4-ball:

$\hat{B}=B\cup\bigcup_{b\in\mathbb{b}}h^{(b)}(I^{4}).$

Let us show that
$(\partial\hat{B},A\cap\partial\hat{B})\approx(\mathbb{S}^{3},L_{\mathbb{b}})$.
Indeed, when the 4-cube $h^{(b)}(I^{4})$ is attached to $B$, the following
surgery is applied to the boundary link: the 3-ball $h^{(b)}(I^{4}\times 0)$,
which is a face of the attached 4-ball, is replaced by the union of the seven
other faces. One can see in Figure 4 that this operation coincides with the
band-move on $L$ along $b$ (cf. Figure 1). ∎

Figure 4 Replacement of one face of $I^{4}$ by seven other faces.

######  Corollary \corConstr

If $\mathbb{b}^{\prime}=\mathbb{b}^{\prime}_{0}\cup\mathbb{b}^{\prime}_{1}$
where $\mathbb{b}^{\prime}_{0}$ is a trivial subset of $\mathbb{b}^{\prime}$
and each band in $\mathbb{b}^{\prime}_{1}$ is positive $\theta$-monotone, then
$L_{\mathbb{b}}$ is a $\mathbb{C}$-boundary.

###### Demonstration Proof

We continue the sequences $(L_{1},\dots\;)$ and $(\mathbb{b}_{1},\dots\;)$ as
shown in Figure 5. Then each positive $\theta$-monotone band is transformed
into a trivial band. ∎

Figure 5 From a positive $\theta$-monotone band to a trivial band.

Example 2.3. It is almost evident (see [7, §2]) that $L\sqcup-L^{*}$ and
$L\,\\#\,{-L^{*}}$ are $\mathbb{C}$-boundaries, if $L$ is a quasipositive link
(in $L\,\\#\,{-L^{*}}$, some component of $L$ is supposed to be joint with its
mirror image). This fact can be also obtained as a simplest application of
Proposition 2.1. Indeed, let $\beta$ be a quasipositive braid whose braid
closure is $L$. We start with the geometric braid with bands shown in Figure
6, and apply positive $\theta$-monotone band-moves to the $\beta^{-1}$ half so
that it is transformed into a trivial braid.

Figure 6 $\mathbb{C}$-boundary realization of $L\,\sqcup-L^{*}$ (on the left)
and $L\,\\#\,{-L^{*}}$ (on the right) based on Proposition 2.1.

Example 2.4. Similarly to Example 2.3, we obtain a $\mathbb{C}$-boundary
realization of $L\,\sqcup-L_{1}^{*}$ and $L\,\\#\,{-L_{1}^{*}}$ where $L$ and
$L_{1}$ are the braid closures of $\beta=\beta_{1}\beta_{2}$ and $\beta_{1}$
respectively for quasipositive braids $\beta_{1}$ and $\beta_{2}$: just
replace $\beta^{-1}$ by $\beta_{1}^{-1}$ in Figure 6.

Example 2.5. Similarly to the previous examples, we obtain a
$\mathbb{C}$-boundary realization of $L\,\\#_{(K,-K^{*})}\,{-L^{*}}$ where $L$
is the braid closure of an $n$-braid $\beta$ belonging to the monoid $M$
generated by
$\\{\sigma_{1},\dots,\sigma_{k-1},\sigma_{k}^{-1},\dots,\sigma_{n-1}^{-1}\\}$
for some $k\leq n$, and $K$ is the component of $L$ containing the $k$-th
string of $\beta$. To this end, we start with a configuration as in Figure 6
but with bands attached to all the strings except the $k$-th one. Then, by a
sequence of positive $\theta$-monotone band-moves we eliminate all the
negative crossings and obtain a trivial collection of bands attached to a
positive braid. In particular $4_{1}\\#4_{1}$ is a $\mathbb{C}$-boundary:
$4_{1}$ is the braid closure of $(\sigma_{1}\sigma_{2}^{-1})^{2}$.

The same construction (and its generalization in the spirit of Example 2.4)
also applies if $M$ is replaced by the monoid generated by

$\\{a\sigma_{1}a^{-1}\mid
a\in\iota_{0}(B_{k})\\}\cup\\{a\sigma_{k}^{-1}a^{-1}\mid
a\in\iota_{k-1}(B_{n-k+1})\\}$

where $\iota_{p}:B_{m}\to B_{n}$ (for $m+p\leq n$) is the homomorhism of the
braid groups which takes $\sigma_{i}$ of $B_{m}$ to $\sigma_{i+p}$ of $B_{n}$.

### 2.3. When the obtained $\mathbb{C}$-boundary is strong?

Let the setting be as in §2.2. Suppose that $\mathbb{b}^{\prime}$ is trivial,
and hence $L_{\mathbb{b}}$ is a $\mathbb{C}$-boundary by Proposition 2.1. Let
us explain how to check if it is strong or not.

Let $G$ be the graph whose vertices correspond to all components (which are
open intervals or circles) of $L_{i}\setminus\mathbb{b}_{i}$ for all
$i=0,\dots,n$, and the edges are defined as follows. If
$L_{i}\cup\mathbb{b}_{i}$ is obtained from $L_{i-1}\cup\mathbb{b}_{i-1}$ by a
$\theta$-monotone isotopy, then the corresponding pairs of vertices are
connected by an edge of $G$. If $L_{i}=(L_{i-1})_{c_{i}}$ and
$\mathbb{b}_{i}=\mathbb{b}_{i-1}$, then all the four vertices corresponding
the components of $L_{i-1}\setminus\mathbb{b}_{i-1}$ and
$L_{i}\setminus\mathbb{b}_{i}$ adjacent to $c_{i}$ are connected to each other
by edges of $G$.

######  Proposition \propConstrStrong

If each vertex of $G$ corresponding to a vanishing arc of
$(L^{\prime},\mathbb{b}^{\prime})$ is connected by a path with a vertex
corresponding to a non-vanishing arc of $(L^{\prime},\mathbb{b}^{\prime})$,
then the link $L_{\mathbb{b}}$ is a strong $\mathbb{C}$-boundary.

Note that the hypothesis of Proposition 2.6 is equivalent to the fact that the
sequences $L_{1},\dots,L_{n}$ and $\mathbb{b}_{1},\dots,\mathbb{b}_{n}$ can be
extended so that the graph $G$ becomes connected. One can check that the
hypothesis of Proposition 2.6 does not hold for $L\sqcup-L^{*}$ in Example 2.3
(cf. Question 4.5 below).

Example 2.7. (A new $\mathbb{C}$-boundary realization of $5_{1}^{2}$.) In [7,
Prop. 6.3] we proved that the link $5_{1}^{2}$ (the mirror image of L5a1 in
[9]) is a strong $\mathbb{C}$-boundary. Corollary 2.2 provides another proof
which seems to be simpler (see Figure 7).

Figure 7 A new strong $\mathbb{C}$-boundary realization of $5_{1}^{2}$.

## 3\. Links with six crossings

As in [7], we denote the class of quasipositive links, the class of
$\mathbb{C}$-boundaries, and the class of strong $\mathbb{C}$-boundaries by
$\Cal{Q}$, $\Cal{B}$, and $\Cal{SB}$ respectively. In this section, for each
link with crossing number 6, we determine if it belongs to these classes (with
two exceptions for the class $\Cal{SB}$). In Tables 1–3 we give the answers
for all such links which do not have an unknot split component, however, the
answers for links of the form $L\sqcup O\sqcup\dots\sqcup O$ with $6$
crossings easily follow (see [7, §6]). More explanations about Tables 1–3 are
given in §3.3.

Table 1. Prime links.

$L$ braid $\operatorname{sgn}\operatorname{lk}(,)$ $L\in\Cal{Q}$
$L\in\Cal{SB}$ $L\in\Cal{B}$ $\chi_{s}(L)$ $\chi_{s}^{-}(L)$ §3.3 $6_{1}$
$112\bar{1}\bar{3}2\bar{3}$ no yes yes F $6_{1}^{*}$ no yes yes F $6_{2}$
$111\bar{2}1\bar{2}$ no yes yes F $6_{2}^{*}$ no no no $-1$ 1 B $6_{3}$
$11\bar{2}1\bar{2}\bar{2}$ no no no $-1$ 1 B L6a1(0) $-$ no no no $-2$ 2 B
L6a1(0)∗ $\bar{1}2\bar{3}2\bar{1}232$ $+$ no no no $-2$ 0 B L6a1(1)
$112\bar{1}211$ $+$ yes yes yes L6a1(1)∗ $-$ no no no $-2$ 2 B L6a2(0) $-$ no
no no $-2$ 2 B L6a2(0)∗ $1\bar{2}12211$ $+$ yes yes yes L6a3(0) $-$ no no no
$-4$ 2 B L6a3(0)∗ $111111$ $+$ yes yes yes L6a3(1)
$2\bar{3}2\bar{1}2\bar{3}2331$ $+$ yes yes yes L6a3(1)∗ $-$ no no no $0$ 2 B
L6a4 $1\bar{2}1\bar{2}1\bar{2}$ 0 0 0 no yes yes F L6a5(0,0) $---$ no no no
$-1$ 3 C L6a5(0,0)∗ $\bar{1}\bar{2}321213\bar{2}32$ $+++$ yes yes yes
L6a5(0,1) $1\bar{2}11\bar{2}1$ $++-$ no yes yes F L6a5(0,1)∗ $--+$ no no no
$-1$ 1 C L6n1(0,0) $\bar{1}\bar{1}2112$ $++-$ yes yes yes L6n1(0,0)∗ $--+$ no
? yes [7] L6n1(0,1) $---$ no no no $-3$ 3 C L6n1(0,1)∗ $121212$ $+++$ yes yes
yes

Table 2. Composite non-split links.

$L$ braid $L\in\Cal{Q}$ $L\in\Cal{SB}$ $L\in\Cal{B}$ $\chi_{s}(L)$
$\chi_{s}^{-}(L)$ see §3.3 $3_{1}\\#3_{1}$ $111222$ yes yes yes
$3_{1}\\#3_{1}^{*}$ no yes yes E $3_{1}^{*}\\#3_{1}^{*}$ no no no $-3$ 1 C
$4_{1}\\#2_{1}$ $1\bar{2}1\bar{2}33$ no yes yes F $4_{1}\\#2_{1}^{*}$
$1\bar{2}1\bar{2}\bar{3}\bar{3}$ no no no 0 2 B L4a1(0)$\\#2_{1}\;$ no no no
$-1$ 1 B L4a1(0)∗$\\#2_{1}$ $2\bar{1}21133$ yes yes yes
L4a1(0)$\\#2_{1}^{*}\;$ no no no $-1$ 3 B L4a1(0)∗$\\#2_{1}^{*}$
$2\bar{1}211\bar{3}\bar{3}$ no no no $-1$ 1 B L4a1(1)$\\#2_{1}\;$ $111122$ yes
yes yes L4a1(1)∗$\\#2_{1}$ no no no $-1$ 1 B L4a1(1)$\\#2_{1}^{*}\;$
$1111\bar{2}\bar{2}$ no yes yes E L4a1(1)∗$\\#2_{1}^{*}$ no no no $-1$ 3 B
$I_{+++}$ $112233$ yes yes yes $I_{++-}$ $1122\bar{3}\bar{3}$ no yes yes A
$I_{+-+}$ $11\bar{2}\bar{2}33$ no yes yes A,F $I_{+--}$
$11\bar{2}\bar{2}\bar{3}\bar{3}$ no no no $0$ 2 C $I_{-+-}$
$\bar{1}\bar{1}22\bar{3}\bar{3}$ no no no $0$ 2 C $I_{---}$
$\bar{1}\bar{1}\bar{2}\bar{2}\bar{3}\bar{3}$ no no no $-2$ 4 B,C $Y_{+++}$
$1122322\bar{3}$ yes yes yes $Y_{++-}$ $11223\bar{2}\bar{2}\bar{3}$ no yes yes
$0$ 2 A $Y_{+--}$ $11\bar{2}\bar{2}3\bar{2}\bar{2}\bar{3}$ no no no $0$ 2 C
$Y_{---}$ $\bar{1}\bar{1}\bar{2}\bar{2}3\bar{2}\bar{2}\bar{3}$ no no no $-2$ 4
B,C

Table 3. Split links.

$L$ braid $L\in\Cal{Q}$ $L\in\Cal{SB}$ $L\in\Cal{B}$ $\chi_{s}(L)$
$\chi_{s}^{-}(L)$ see §3.3 $3_{1}\sqcup 3_{1}$ $111\,333$ yes yes yes
$3_{1}\sqcup 3_{1}^{*}$ no ? yes E $3_{1}^{*}\sqcup 3_{1}^{*}$ no no no $-2$ 2
C $4_{1}\sqcup 2_{1}$ $1\bar{2}1\bar{2}\,44$ no no no $-1$ 1 D $4_{1}\sqcup
2_{1}^{*}$ $1\bar{2}1\bar{2}\,\bar{4}\bar{4}$ no no no $-1$ 3 D
L4a1(0)${}\sqcup 2_{1}\;$ no no no 0 2 D L4a1(0)∗${}\sqcup 2_{1}$
$2\bar{1}211\,44$ yes yes yes L4a1(0)${}\sqcup 2_{1}^{*}\;$ no no no 0 4 C
L4a1(0)∗${}\sqcup 2_{1}^{*}$ $2\bar{1}211\,\bar{4}\bar{4}$ no no no 0 2 D
L4a1(1)${}\sqcup 2_{1}\;$ $1111\,33$ yes yes yes L4a1(1)∗${}\sqcup 2_{1}$ no
no no $-2$ 0 D L4a1(1)${}\sqcup 2_{1}^{*}\;$ $1111\,\bar{3}\bar{3}$ no no yes
$-2$ 0 E L4a1(1)∗${}\sqcup 2_{1}^{*}$ no no no $-2$ 2 C $2_{1}\\#2_{1}\sqcup
2_{1}\;$ $1122\,44$ yes yes yes $2_{1}\\#2_{1}^{*}\sqcup 2_{1}\;$
$11\bar{2}\bar{2}\,44$ no yes yes A $2_{1}^{*}\\#2_{1}^{*}\sqcup 2_{1}\;$
$\bar{1}\bar{1}\bar{2}\bar{2}\,44$ no no no $-1$ 3 D $2_{1}\\#2_{1}\sqcup
2_{1}^{*}\;$ $1122\,\bar{4}\bar{4}$ no no yes $-1$ 1 A
$2_{1}\\#2_{1}^{*}\sqcup 2_{1}^{*}\;$ $11\bar{2}\bar{2}\,\bar{4}\bar{4}$ no no
no $1$ 3 C $2_{1}^{*}\\#2_{1}^{*}\sqcup 2_{1}^{*}\;$
$\bar{1}\bar{1}\bar{2}\bar{2}\,\bar{4}\bar{4}$ no no no $-1$ 5 C $2_{1}\sqcup
2_{1}\sqcup 2_{1}\;$ $11\,33\,55$ yes yes yes $2_{1}\sqcup 2_{1}\sqcup
2_{1}^{*}\;$ $11\,33\,\bar{5}\bar{5}$ no no yes 0 2 A $2_{1}\sqcup
2_{1}^{*}\sqcup 2_{1}^{*}\;$ $11\,\bar{3}\bar{3}\,\bar{5}\bar{5}$ no no no 0 4
D $2_{1}^{*}\sqcup 2_{1}^{*}\sqcup 2_{1}^{*}\;$
$\bar{1}\bar{1}\,\bar{3}\bar{3}\,\bar{5}\bar{5}$ no no no 0 6 C

### 3.1. The links $2_{1}\\#2_{1}\\#2_{1}$, $2_{1}\\#2_{1}\\#2_{1}^{*}$, etc

Among the links with 6 crossings, almost all connected sums are uniquely
determined by their summands. The only exceptions are connected sums of three
copies of the Hopf link $2_{1}$ and/or its mirror image $2_{1}^{*}$. Such a
sum is determined by the following graph with signed edges. Its vertices
correspond to the link components. Two vertices are connected with an edge if
the corresponding components are linked, and this edge is labeled by the
linking number, which is always $\pm 1$. This graph is a tree. There are only
two 4-vertex trees: an $I$-shaped tree (a 4-path) and a $Y$-shaped tree (with
a vertex of degree 3). We denote the corresponding links by $I_{\pm\pm\pm}$
and $Y_{\pm\pm\pm}$ respectively. The signs in the notation $I_{\pm\pm\pm}$
appear in the same order as in the graph.

Any connected sum of two strong $\mathbb{C}$-boundaries is a strong
$\mathbb{C}$-boundary, and $2_{1}\\#2_{1}^{*}$ is a strong
$\mathbb{C}$-boundary (see Example 2.3). Hence all links of the form
$2_{1}\\#2_{1}\\#2_{1}^{*}$ (i.e., $I_{++-}$, $I_{+-+}$, and $Y_{++-}$) are
strong $\mathbb{C}$-boundaries. Note that $I_{++-}$ and $Y_{++-}$ are strong
$\mathbb{C}$-boundaries by [7, Theorem 2.1], and we give another strong
$\mathbb{C}$-boundary realization of $I_{+-+}$ in Figure 10.

### 3.2. Restrictions on $\mathbb{C}$-boundaries

Our proofs that some links are not (strong) $\mathbb{C}$-boundaries are based
on Kronheimer-Mrowka’s Theorem [6] (former Thom Conjecture) and its version
for immersed 2-surfaces in $\mathbb{C}P^{2}$ with negative double points (see
[4], [10, §2]), which was actually proven in [6] but was not explicitly
formulated.

######  Theorem \thKM

(The Immersed Thom Conjecture.) Let $\Sigma$ be a connected oriented closed
surface of genus $g$ and $j:\Sigma\to\mathbb{C}P^{2}$ be an immersion which
has only negative ordinary double points as self-crossings. Let
$j_{*}([\Sigma])=d[\mathbb{C}P^{1}]\in H_{2}(\mathbb{C}P^{2})$ with $d>0$.
Then $g$ is bounded below by the genus of a smooth algebraic curve of degree
$d$, that is, $g\geq(d-1)(d-2)/2$.

Given a link $L$ in $S^{3}=\partial B^{4}$, we define the slice Euler
characteristic of $L$ by $\chi_{s}(L)=\max_{\Sigma}\chi(\Sigma)$ where the
maximum is taken over all embedded smooth oriented surfaces $\Sigma$ without
closed components, such that $\partial\Sigma=L$. Similarly, we define the
slice negatively immersed Euler characteristic of $L$ by
$\chi_{s}^{-}(L)=\max_{(\Sigma,j)}\chi(\Sigma)$ where the maximum is taken
over all immersions $j:(\Sigma,\partial\Sigma)\to(B^{4},S^{3})$ of oriented
surfaces $\Sigma$ without closed components such that $j(\Sigma)$ has only
negative double points and $j(\partial\Sigma)=L$. Theorem 3.1 easily implies
the following two statements.

######  Proposition \propB

(See [7, Prop. 3.2 and Lem. 3.8].) If $\chi_{s}(L)<\chi_{s}^{-}(L)$, then
$L\not\in\Cal{SB}$. If, moreover, $L$ does not have a proper sublink which has
zero linking number with its complement (for example, if $L$ is a knot), then
$L\not\in\Cal{B}$.

We see in [7, Table 1] and in Tables 1–3 here that for any link with $\leq 6$
crossings with a single exception $3_{1}^{*}\\#2_{1}$, we have either
$L\in\Cal{SB}$ or $\chi_{s}(L)<\chi_{s}^{-}(L)$.

######  Proposition \propC

(See [7, Prop. 3.7]; cf. [2, Cor. 1.6].) If $L$ is a strong
$\mathbb{C}$-boundary and $-L^{*}$ is a (not necessarily strong)
$\mathbb{C}$-boundary, then $\chi_{s}(L)=\chi_{s}^{-}(L)\geq 1$.

Now let us treat the cases which are not covered by Propositions 3.2, 3.3.

######  Proposition \propD

The links marked by “D” in the last column of Table 3 are not
$\mathbb{C}$-boundaries.

###### Demonstration Proof

Case 1. The links $4_{1}\sqcup 2_{1}$ and $4_{1}\sqcup 2_{1}^{*}$. Suppose
that there exists a $\mathbb{C}$-boundary realization $L=A\cap B$ of one of
these links. Let $L=L_{4}\cup L_{2}$ where $L_{4}$ is $4_{1}$ and $L_{2}$ is
$2_{1}$ or $2_{1}^{*}$. $L$ is not a strong $\mathbb{C}$-boundary by
Proposition 3.2. Hence $A\setminus B$ is a disjoint union $A_{2}\cup A_{4}$
with $\partial A_{k}=-L_{k}$, where either $A_{4}$ or $A_{2}$ is bounded. The
interior piece $A_{0}=A\cap B$ of $A$ is connected because otherwise $A$ would
be disconnected (here we use the fact that there is only one partition of $L$
into sublinks with zero linking number). Hence $\chi(A_{0})\leq-1$. Let us
replace $A_{0}$ by $A^{\prime}_{2}\sqcup A^{\prime}_{4}$, $\partial
A^{\prime}_{k}=L_{k}$, where $A^{\prime}_{2}$ is an embedded annulus and
$A^{\prime}_{4}$ is an immersed disk with a negative self-crossing. Recall
that $A_{2}$ or $A_{4}$ is bounded. If it is $A_{2}$, then $A_{4}\cup
A^{\prime}_{4}$ contradicts Theorem 3.1; if it is $A_{4}$, then so does
$A_{2}\cup A^{\prime}_{2}$ because in this case $\chi(A_{4})\leq-1$ ($4_{1}$
is not slice), and hence the replacement $A_{0}\cup A_{4}\to A^{\prime}_{2}$
decreases $\chi_{s}^{-}$ (see Figure 8).

Figure 8 Proof that $4_{1}\sqcup 2_{1}$ and $4_{1}\sqcup 2_{1}^{*}$ are not
$\mathbb{C}$-boundaries.

Case 2. The links L4a1(0)${}\sqcup 2_{1}$, L4a1(0)${}^{*}\sqcup 2_{1}^{*}$,
and L4a1(1)${}^{*}\sqcup 2_{1}$. The same proof as in the previous case; see
Figure 9.

Figure 9 Proof that L4a1(0)${}\sqcup 2_{1}$, L4a1(0)${}^{*}\sqcup 2_{1}^{*}$,
L4a1(1)${}^{*}\sqcup 2_{1}\not\in\Cal{B}$.

Case 3. The link $L=2_{1}^{*}\\#2_{1}^{*}\sqcup 2_{1}$. Suppose that $L=A\cap
B$ is a $\mathbb{C}$-boundary realization of $L$. We have $\chi_{s}(L)=-1$ and
$\chi_{s}^{-}(L)=3$. Thus, if we replace $A\cap B$ with a negatively self-
crossing surface $A^{\prime}_{0}$ which realizes $\chi_{s}^{-}(L)=3$, we
obtain a contradiction with Theorem 3.1 unless at least two spheres split out.

Note that if we glue together several surfaces along their boundaries, we may
obtain a sphere only if at least two of the glued surfaces are disks. The link
$L$ does not have any component which has zero linking number with its
complement, hence no component of $A\setminus B$ is a disk. The negatively
self-crossing surface $A^{\prime}_{0}$ has at most three disks, hence at most
one sphere can split out.

Case 4. The link $L=2_{1}\sqcup 2_{1}^{*}\sqcup 2_{1}^{*}$. Suppose that
$L=A\cap B$ is a $\mathbb{C}$-boundary realization of $L$. We have
$\chi_{s}(L)=0$ and $\chi_{s}^{-}(L)=4$. Thus, as in Case 3, if we replace
$A\cap B$ with a negatively self-crossing surface $A^{\prime}_{0}$, at least
two spheres must split out. Again as in Case 3, no component of $A\setminus B$
is a disk but this time $A^{\prime}_{0}$ may have four disks (bounded by the
components of $2_{1}^{*}\sqcup 2_{1}^{*}$). Thus all these disks are present
in $A^{\prime}_{0}$, all of them make part of the two spheres which split out,
and $\chi(A\cap B)=0$ (if $\chi(A\cap B)$ were negative, then one more sphere
should split out). This may happen only when $A\setminus B=A_{1}^{-}\cup
A_{2}^{-}\cup A^{+}$ where $A_{j}^{-}$ are cylinders, $A^{+}$ is the unbounded
component, each $\partial A_{j}^{-}$ is $2_{1}^{*}$, and $\partial A^{+}$ is
$2_{1}$.

The condition $\chi(A\cap B)=0$ implies that $A\cap B$ is a union of three
cylinders. Since all the linking numbers between their boundaries are zero,
one of them must be bounded by $2_{1}^{*}$ (while a priori the boundaries of
the other two may realize any $(2,2)$-partition of the four components of
$2_{1}\sqcup 2_{1}^{*}$). This fact contradicts the connectedness of $A$. ∎

$\,\,\,6_{1}$

$\,\,\,6_{1}^{*}$

$\,\,\,6_{2}$

$4_{1}\\#2_{1}$

L6a4

L6a5(0,1)

$I_{+-+}$

Figure 10 A strong $\mathbb{C}$-boundary realization of the knots $6_{1}$,
$6_{1}^{*}$, $6_{2}$, $4_{1}\\#2_{1}$ and the links L6a4 (Borromeo rings),
L6a5(0,1), and $I_{+-+}$.

### 3.3. Comments on Tables 1–3

The tables are organized similarly to Table 1 in [7]. The links are named
according to [8], [9] except the two-component Hopf links $2_{1}$,
$2_{1}^{*}$, and their connected sums, whose notation is explained in §3.1.
The braid notation and the signs of linking numbers help to identify the
links. The invariants $\chi_{s}(L)$ and $\chi_{s}^{-}(L)$ are introduced in
§3.2. They are computed by standard methods (some of them are mentioned in [7,
§6]) and we omit the details. The same for the quasipositivity: in all the
considered cases, if $L\in\Cal{Q}$, this fact is clear from the braid given in
the tables, and if $L\not\in\Cal{Q}$, this fact follows either from [12], or
from the Franks-Williams-Morton Inequality [5], [11] in the form given in [2,
Thm. 3.2] (see [7, Thm. 6.1]). In all the cases when
$L\in\Cal{B}\setminus\Cal{SB}$, the fact that $L\not\in\Cal{SB}$ follows from
the inequality $\chi_{s}(L)<\chi_{s}^{-}(L)$ (see Proposition 3.2).

The letters A–F in the last column of each table mean the following:

## 4\. Concluding remarks

### 4.1. Squeezed knots

The referee drew my attention to a very interesting paper [3] by Feller,
Lewark, and Lobb, where they introduce squeezed knots. These are knots which
appear as a slice of a genus-minimizing oriented connected smooth cobordism
between a positive torus knot and a negative torus knot.

######  Proposition \propSqueezed

Any $\mathbb{C}$-boundary knot is squeezed.

###### Demonstration Proof

Let $A\cap B$, $A=\\{f(z,w)=0\\}$, be a $\mathbb{C}$-boundary realization of a
knot $K$. Let $n=\deg f$. Without loss of generality we may assume that $A$ is
smooth, $f(0,0)=0$, and $\deg_{z}f=n$. Let
$A_{\varepsilon}=\\{f(z,w)=\varepsilon w^{n+1}\\}$. If $|\varepsilon|$ is
sufficiently small, then $K_{\varepsilon}=A_{\varepsilon}\cap B$ is isotopic
to $K$ and $A_{\varepsilon}\cap(B_{R}\setminus B_{r})$, $0<r\ll 1\ll R$, is a
cobordism between the positive torus knot $T(n,n+1)$ and the unknot. This
cobordism is genus-minimizing by Kronheimer-Mrowka’s theorem. ∎

As proven in [3], the knots $9_{42}$, $10_{125}$, $10_{132}$, and $10_{136}$
are not squeezed. Hence they and their mirrors are not
$\mathbb{C}$-boundaries. For these knots themselves, this fact follows from
Proposition 3.2, because they are not slice (see [8]) but $\chi_{s}^{-}=1$.
The latter fact can be checked using the braids (see the beginning of §6 in
[7]):

$\begin{split}\chi_{s}^{-}(9_{42})&=\chi_{s}^{-}(\bar{1}\bar{1}\bar{1}211\bar{3}2\bar{3})\geq\chi_{s}(\bar{1}(\bar{1}\bar{1}211)(\bar{3}23))=1,\\\
\chi_{s}^{-}(10_{125})&=\chi_{s}^{-}(\bar{1}\bar{1}\bar{1}\bar{1}\bar{1}21112)\geq\chi_{s}((\bar{1}\bar{1}\bar{1}2111)2)=1,\\\
\chi_{s}^{-}(10_{132})&=\chi_{s}^{-}(111\bar{2}\bar{1}\bar{1}\bar{2}\bar{3}2\bar{3}\bar{3})\geq\chi_{s}(1(11\bar{2}\bar{1}\bar{1})(\bar{2}\bar{3}2))=1,\\\
\chi_{s}^{-}(10_{136})&=\chi_{s}^{-}(\bar{1}\bar{1}\bar{2}3\bar{2}1\bar{2}\bar{2}322)\geq\chi_{s}((23\bar{2})1(\bar{2}\bar{2}322))=1.\end{split}$

In contrary, the methods of our paper are not sufficient to prove that the
mirror images of these four knots are not $\mathbb{C}$-boundaries. In the
first arxiv version of this paper I asked whether
$\chi_{s}^{-}(K)=\chi_{s}(K)$ implies that $K$ is $\mathbb{C}$-boundary. The
knots $9_{42}^{*}$, $10_{125}*$, $10_{136}^{*}$ provide a negative answer.
Indeed, if $K$ is one of them, then the signature of $K\\#2_{1}$ equals $3$
which implies that $\chi_{s}^{-}(K)<0$, hence
$\chi_{s}^{-}(K)=\chi_{s}(K)=-1$.

Notice that a negative answer to the same question for links is provided by
the link $L=3_{1}^{*}\sqcup 2_{1}$. We have $\chi_{s}(L)=\chi_{s}^{-}(L)=0$
whereas $L\not\in\Cal{B}$ by Proposition 3.3 (see [7]). It also seems
plausible that $\chi_{s}^{-}(K)=\chi_{s}(K)=-1$ for $K=8_{18}$ (the braid
closure of $(\sigma_{1}\sigma_{2}^{-1})^{4}$), whereas $K=K^{*}$ and hence $K$
is not a $\mathbb{C}$-boundary by [2, Cor. 1.6].

### 4.2. Slice knots

Up to now, all known proofs that a certain knot is not $\mathbb{C}$-boundary,
automatically gives that all concordant knot are not $\mathbb{C}$-boundaries
neither. Thus there are no tools to prove that a slice knot is not a
$\mathbb{C}$-boundary.

There are 15 slice knots with $\leq 9$ crossings: $6_{1}$, $8_{8}$, $8_{9}$,
$8_{20}$, $9_{27}$, $9_{41}$, $9_{46}$, their mirror images ($8_{9}$ is
amphicheiral), $3_{1}\\#\,3_{1}^{*}$, and $4_{1}\\#\,4_{1}$. Among them, the
following ten are realized as $\mathbb{C}$-boundaries: $6_{1}$, $6_{1}^{*}$
(Figure 10); $8_{8}$, $8_{9}$, $9_{27}^{*}$, $9_{41}^{*}$ (Figure 11);
$8_{20}^{*}$, $9_{46}^{*}$ (quasipositive knots); $3_{1}\\#\,3_{1}^{*}$ and
$4_{1}\\#\,4_{1}$ (Examples 2.3 and 2.5). The realizability of $8_{8}^{*}$,
$8_{20}$, $9_{27}$, $9_{41}$, and $9_{46}$ is unknown.

$8_{8}$

$8_{9}$

$9_{27}^{*}$

$9_{41}^{*}$

Figure 11 A $\mathbb{C}$-boundary realization of the knots $8_{8}$,
$8_{9}=8_{9}^{*}$, $9_{27}^{*}$, $9_{41}^{*}$.

### 4.3. Some open questions

The study of $\mathbb{C}$-boundary links is at the very beginning. So, most
questions on this subject are open. However, I would like to emphasize some of
them.

Question 4.2. Can any $\mathbb{C}$-boundary be realized by the construction
from §2?

Question 4.3. Do there exist slice knots which are not
$\mathbb{C}$-boundaries? In particular, does there exist a knot $K$ such that
$K\,\\#\,{-K^{*}}$ is not a $\mathbb{C}$-boundary?

As we pointed out in §4.2, if the answer is affirmative, then essentially new
ideas should be evoked to prove this fact.

Question 4.4. Is it true that for any slice knot $K$, either $K$ or $K^{*}$ is
$\mathbb{C}$-boundary?

The answer is affirmative for knots with at most 9 crossings (see §4.2).

Question 4.5. Is it true that $K\sqcup-K^{*}$ is not a strong
$\mathbb{C}$-boundary for any non-trivial quasipositive knot $K$?

Question 4.6. (Cancellation of unknot in a split sum.) If $L\sqcup O$ is a
(strong) $\mathbb{C}$-boundary, is $L$ a (strong) $\mathbb{C}$-boundary?

Question 4.7. Do there exist non-split non-strong $\mathbb{C}$-boundaries?

The link L6n1(0,0)∗ (see [7, §4]) is a candidate for an affirmative answer.

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