# Guts in Sutured Decompositions and the Thurston Norm

Ian Agol  University of California, Berkeley<EMAIL_ADDRESS>and
Yue Zhang  University of California, Berkeley<EMAIL_ADDRESS>
###### Abstract.

We construct an invariant called guts for second homology classes in
irreducible 3-manifolds with toral boundary and non-degenerate Thurston norm.
We prove that the guts of second homology classes in each Thurston cone are
invariant under a natural condition. We show that the guts of different
homology classes are related by sutured decompositions. As an application, an
invariant of knot complements is given and is computed in a few interesting
cases. Besides, we show that the dimension of a maximal simplex in a Kakimizu
Complex is an invariant.

##  1\. Introduction

Thurston [Thu86] defined a norm on the homology $H_{2}(M,\partial M)$ called
the Thurston norm for a compact 3-manifold $M$, which is an important
invariant of 3-manifolds. Gabai [Gab83] generalized the definition and defined
the notion of sutured manifolds which is related to foliations of manifolds.
Furthermore, one can decompose a sutured manifold into two submanifolds, the
windows and the guts, specializing the notion of guts defined by Gabai and
Kazez for laminations [GK98]. Guts can be used to understand how close a
manifold is to a fibration. Agol [Ago08] and later Friedl and Kitayama [FK14]
used guts to show that an irreducible 3-manifold with fundamental group that
satisfies a certain group-theoretic property called $RFRS$ is virtually
fibered.

Guts is also defined for pared manifolds. It reveals information about the
volume of a hyperbolic manifold. Agol, Storm and Thurston [ADST05] gave an
estimate of the volume of a manifold from the volume of guts related to
incompressible surfaces. Agol [Ago10] and Yoshida [Yos13] found the minimal
volumes of 2-cusped and 4-cusped orientable hyperbolic 3-manifolds
respectively with the help of estimating the volume of guts.

In this paper, we give a definition for a _facet surface_ of a primitive
homology class in an irreducible 3-manifold as a maximal collection of
disjointly embedded Thurston norm minimizing surfaces for that homology class
and define the _guts_ of this homology class as the nontrivial parts in the
sutured decomposition of the 3-manifold along a facet surface.

In the following, we consider irreducible, orientable manifolds with torus
boundary components and non-degenerate Thurston norm.

We first show that the guts for homology classes are well defined.

######  Theorem 1.1.

Let $M$ be an irreducible, orientable $3$-manifold with boundary a disjoint
union of tori $\sqcup_{i=1}^{n}P_{i}$ and non-degenerate Thurston norm. The
guts $\Gamma(M,F)$ for facet surfaces representing a primitive element $z$ in
$H_{2}(M,\partial M;\mathbb{Z})$ do not depend on the selection of facet
surfaces and product decomposition surfaces up to the equivalence of guts.

By constructing branched surfaces, we prove that the guts are invariant for
second homology classes in each Thurston cone under a natural condition (see
Definition 4.1). This is proved by using a nice property of double curve sum
operations [Mil19, Proposition 2.8].

######  Theorem 1.2.

Let $M$ be an irreducible, orientable $3$-manifold with boundary a disjoint
union of tori $\sqcup_{i=1}^{n}P_{i}$ and non-degenerate Thurston norm. Let
$y,z$ be two elements in an open face of the Thurston sphere. If there is an
open segment $(v,w)$ containing $y,z$ in the open face such that the
restrictions of $v$ and $w$ on each boundary component are not in opposite
orientations, the guts $\Gamma(y)$ is equivalent to $\Gamma(z)$.

Agol’s criteria for virtual fibering is based on a key finding [Ago08, Lemma
4.1] and we state a generalization of it.

######  Theorem 1.3.

Let $z$ lie on a $k$-codimensional open Thurston cone $\Delta$. We denote the
subspace spanned by $\Delta$ as $V$. Then the kernel of the restriction map
$\varphi:H_{2}(M,\partial M;\mathbb{Z})\to
H_{2}(\Gamma(z),\partial\Gamma(z);\mathbb{Z})$ is a subspace of $V$.

If there is a neighborhood of $z$ in $\Delta$ which does not contain two
elements whose restrictions on a boundary component are in opposite
orientations, the kernel of $\varphi$ is exactly $V$ and the rank of the image
of $\varphi$ is $k$, i.e. the rank of the image of

$H^{1}(M)\to H^{1}(\Gamma(z))$

is $k$.

The first part of this theorem is proved by using the canonical foliation on
the complement of the guts. We prove the opposite direction via double curve
sum operations.

With the help of the preceding theorem, we show that guts of homology classes
are related by sutured decompositions.

######  Theorem 1.4.

Let $M$ be an irreducible connected 3-manifold with toral boundary and non-
degenerate Thurston norm. Let $z$ be an element on a $k$-codimensional $(k>0)$
open Thurston face of the Thurston norm. For any choice of $u$ in any
$k^{\prime}$-codimensional $(k^{\prime}<k)$ open Thurston face whose closure
contains $z$, there exists a $w$ on the open segment $(z,u)$ such that
$\Gamma(z)$ can be nontrivially decomposed along a properly norm-minimizing
surface representing $w$.

As an application, an invariant of knot complements is given as the guts of a
maximal collection of disjoint Seifert surfaces (see Definition 7.4). We find
an example of two 2-bridge knots having different guts but the same Floer knot
homology in Example 7.18.

Motivation.

Why define the notion of guts associated to a sutured manifold or a homology
class? As indicated before, the main theorem is a natural generalization of
Thurston’s fibered face theorem, and a generalization used to understand the
virtual fibering conjecture. But another motivation comes from sutured
manifold theory. When performing sutured manifold decompositions, the most
natural ways one will try are horizontal and vertical decompositions. What the
main theorem implies is that after throwing away the product pieces after a
maximal such collection of decompositions, the remaining sutured manifold is
independent of the choices made. So it gives a weak kind of uniqueness for the
initial stages of a sutured manifold hierarchy. Moreover, the sutured Floer
homology is invariant under horizontal and vertical decompositions, hence it
is natural to look for uniqueness after these operations.

Another reason to consider these notions is to investigate the Kakimizu
complex of isotopy classes of minimal genus surfaces bounding a fixed knot.
One consequence of the main theorem is that the dimension of maximal simplices
of the Kakimizu complex is an invariant of the complex (and hence of the knot
type). A generalization of this statement is proved in Theorem 8.2. Our
terminology facet surface is motivated by these considerations. Moreover,
neighboring maximal simplices correspond to replacing one surface with another
one intersecting it, and performing an exchange on the order of the guts. So
hopefully the results in this paper will give some refined information about
the structure of the Kakimizu complex.

Finally, a motivation for studying these notions comes from trying to find
surfaces in the manifold whose complement has maximal volume guts, in order to
get lower bounds on the volume of the manifold. It turns out that the
structure uncovered in the main theorems is useful for analyzing
configurations of surfaces in manifolds of small volume.

Organization.

In Section 2 we recall some standard facts about the Thurston norm and sutured
manifolds.

In Section 3, we define a facet surface of a second homology class in a
3-manifold as a maximal collection of properly norm-minimizing surfaces and
the guts of this homology class as the nontrivial parts in the sutured
decomposition of the 3-manifold along a facet surface. We show that the guts
are invariant among facet surfaces representing the same homology class.

In Section 4, we show that if two elements in an open face of the Thurston
sphere don’t meet any boundary component in opposite orientation, their guts
are equivalent.

After that, we find a relation between the open Thurston cone $\Delta$ and the
restriction map $\varphi$ from $H_{2}(M,\partial M)$ to
$H_{2}(\Gamma(z),\partial\Gamma(z))$ for an element $z$ in $\Delta$ in Section
5.

With the previous results, we are able to show that guts of second homology
classes can be decomposed by surfaces in the manifold in Section 6.

In Section 7, we give some examples of guts and define an invariant for knot
complements. One can gain intuition of the main theorems from this section
before trying to understand the proofs.

In Section 8, we prove that the dimension of maximal simplices of a Kakimizu
complex is an invariant of the complex.

Acknowledgements.

This research is supported by a Simons Investigator award and by the
Mathematical Sciences Research Institute. We thank Nathan Dunfield for
allowing us to use his drawings. The second author is thankful to Chi Cheuk
Tsang, Yi Liu, Yi Ni, Maggie Miller, Michael Landry, Andras Juhasz, and Jiajun
Wang for helpful comments and conversations. The second author is grateful to
the first author for the guidance.

##  2\. The Thurston norm and Sutured Decompositions

This section reviews some background material on the Thurston norm [Thu86] and
sutured manifolds [Gab83].

We denote $\eta(X)$ as a regular neighborhood of $X$. For a properly embedded
surface $S\subset M$, $\eta(S)$ is an open tubular neighborhood of $S$
homeomorphic to the normal bundle, and we use $M\backslash\backslash S$ to
indicate ${M\backslash\eta(S)}$.

In this paper, we sometimes omit the $\mathbb{Z}$ coefficient in homology and
cohomology groups.

######  Definition 2.1.

Let $S$ be a compact connected orientable surface. Define
$\chi_{-}(S)=\max\\{-\chi(S),0\\}$. For a disconnected compact orientable
surface $S$, let $\chi_{-}(S)$ be the sum of $\chi_{-}$ evaluated on the
connected subsurfaces of $S$.

In [Thu86], Thurston defined a pseudonorm on $H_{2}(M,\partial M;\mathbb{R})$
and $H_{2}(M;\mathbb{R})$, and this was generalized by Gabai [Gab83].

######  Definition 2.2.

Let $M$ be a compact oriented 3-manifold. Let $K$ be a subsurface of $\partial
M$. Let $z\in H_{2}(M,K)$. Define the _Thurston norm_ of $z$ to be

$x(z)=\min\\{\chi_{-}(S)\ |\ (S,\partial S)\subset(M,K),\ \text{and}\ [S]=z\in
H_{2}(M,K)\\}.$

Thurston proved that $x(nz)=nx(z)$, and therefore $x$ may be extended to a
norm $x:H_{2}(M,K;\mathbb{Q})\to\mathbb{Q}$. Then $x$ is extended to
$x:H_{2}(M,K;\mathbb{R})\to\mathbb{R}$ by continuity.

Denote by $B_{x}(M)$ the unit ball of the Thurston norm on $H_{2}(M,\partial
M)$. By [Thu86, Theorem 2], $B_{x}(M)$ is a polyhedron and hence we can
decompose $B_{x}(M)$ into faces of various dimensions. We call $B_{x}(M)$ _the
Thurston ball_ , $\partial B_{x}(M)$ _the Thurston sphere_ , each face of
$\partial B_{x}(M)$ _a Thurston face_ and a cone formed by the origin and a
Thurston face _a Thurston cone_. When we say $z$ is on a Thurston face, we
actually mean a multiple of $z$ is on a Thurston face.

######  Remark 2.3.

When the interior of $M$ is a hyperbolic manifold of finite volume, the
Thurston norm is non-degenerate.

######  Definition 2.4.

Let $M$ be a compact $3$-manifold. Let $S$ be an oriented surface representing
$\alpha\in H_{2}(M,\partial M)$ with $\alpha\neq 0$. We say that $S$ is _norm-
minimizing_ if the following conditions hold:

  * •

$S$ has no null-homologous subset of components in $H_{2}(M,\partial M)$,

  * •

$\chi_{-}(S)=x_{M}(\alpha)$.

When the boundary of a manifold is not just a union of tori, we define a
notion of taut surfaces.

######  Definition 2.5.

Let $M$ be a compact $3$-manifold. An oriented surface $(S,\partial S)$ is
_taut_ if

  * •

$S$ is incompressible and no collection of components is homologously trivial
in $H_{2}(M,\partial M)$ and,

  * •

$S$ has minimal $\chi_{-}$ of all surfaces representing $[S,\partial S]$ in
$H_{2}(M,\eta(\partial S))$, we have $x([S])=\chi_{-}(S)$. Here $\eta(\partial
S)$ means the tubular neighborhood of $\partial S$.

######  Lemma 2.6.

If $P$ is a norm-minimizing (or taut) surface, and $Q$ is a union of
components of $P$, then $Q$ is also a norm-minimizing (or taut) surface.

This follows immediately from the triangle inequality for the Thurston norm.

In this chapter, we use an important tool called double curve sum to analyze
the guts of different homology classes.

######  Definition 2.7.

Let $S$ and $T$ be two properly embedded oriented surfaces which are in
general position. By the standard cut-and-paste technique applied to the
intersection curves of $S$ and $T$, we can turn the immersed surface $S\cup T$
into a properly embedded oriented surface $S\oplus T$. The surface $S\oplus T$
is called the _double curve sum_ of $S$ and $T$.

The following lemma and corollary show some simple but important facts about
double curve sum.

######  Lemma 2.8.

If every component of $S\cap T$ is essential in $S$ and $T$, then
$\chi_{-}(S)+\chi_{-}(T)=\chi_{-}(S\oplus T)$.

###### Proof.

This is because $\chi(S)+\chi(T)=\chi(S\oplus T)$ and $S\oplus T$ does not
have components of positive Euler characteristic. ∎

######  Lemma 2.9.

Let $S_{1}$ and $S_{2}$ be two properly embedded oriented surfaces in general
position such that every component of $S_{1}\cap S_{2}$ is essential in
$S_{1}$ and $S_{2}$. If $S_{1}\oplus S_{2}$ is a norm-minimizing surface, then
$S_{1}$ and $S_{2}$ have the minimal $\chi_{-}$ among all surfaces
representing the same element in the second relative homology group
$H_{2}(M,\partial M)$ respectively.

###### Proof.

This is because of the equality
$\chi_{-}(S_{1})+\chi_{-}(S_{2})=\chi_{-}(S_{1}\oplus S_{2})$ and the triangle
inequality for the Thurston norm. ∎

Gabai in [Gab83] introduced the notion of a sutured manifold and a
decomposition surface.

######  Definition 2.10.

A _sutured manifold_ $(M,\gamma)$ is a compact oriented 3-manifold $M$ with
boundary together with a set $\gamma\subset\partial M$ of pairwise disjoint
annuli $A(\gamma)$ and tori $T(\gamma).$ Furthermore, the interior of each
component of $A(\gamma)$ contains a _suture_ , i.e., a homologically
nontrivial oriented simple closed curve. We denote the union of the sutures by
$s(\gamma)$ or $s(M).$ We say that an annulus component of $A(\gamma)$ is a
_sutured annulus_ and a torus component of $T(\gamma)$ is a _sutured torus_.

Finally every component of $R(\gamma)=\partial M\backslash int(\gamma)$ is
oriented. Define $R_{+}(\gamma)$ (or $R_{-}(\gamma)$) to be those components
of $\partial M\setminus int(\gamma)$ whose normal vectors point out of (into)
$M$. The orientation on $R(\gamma)$ must be coherent with respect to
$s(\gamma),$ i.e., if $\delta$ is a component of $\partial R(\gamma)$ and is
given the boundary orientation, then $\delta$ must represent the same homology
class in $H_{1}(\gamma)$ as some suture.

######  Definition 2.11.

Let $(M,\gamma)$ be a sutured manifold, and $S$ a properly embedded surface in
$M$ such that for every component $\lambda$ of $S\cap\gamma$ one of (1)-(3)
holds:

  1. (1)

$\lambda$ is a properly embedded nonseparating arc in an annular component of
$\gamma$,

  2. (2)

$\lambda$ is a simple closed curve in an annular component $A$ of $\gamma$ in
the same homology class as $A\cap s(\gamma)$,

  3. (3)

$\lambda$ is a homotopically nontrivial curve in a toral component $T$ of
$\gamma$, and if $\delta$ is another component of $T\cap S$, then $\lambda$
and $\delta$ represent the same homology class in $H_{1}(T)$.

Furthermore, no component of $\partial S$ bounds a disc in $R(\gamma)$ and no
component of $S$ is a disc with $\partial D\subset R_{+}\cup R_{-}$. Then we
call $S$ a _decomposition surface_ for $(M,\gamma)$. The orientation of $S$
induces orientations on $S_{\pm}\subset\partial\eta(S)$. We can construct a
suture structure on $M^{\prime}=\overline{M\backslash\eta(S)}$:

$\displaystyle\gamma^{\prime}$ $\displaystyle=$ $\displaystyle(\gamma\cap
M^{\prime})\cup\overline{\eta(S^{\prime}_{+}\cap
R_{-})}\cup\overline{\eta(S^{\prime}_{-}\cap R_{+})}$ $\displaystyle
R^{\prime}_{+}$ $\displaystyle=$ $\displaystyle((R_{+}\cap M^{\prime})\cup
S^{\prime}_{+})\backslash\textnormal{int}\gamma^{\prime}$ $\displaystyle
R^{\prime}_{+}$ $\displaystyle=$ $\displaystyle((R_{-}\cap M^{\prime})\cup
S^{\prime}_{-})\backslash\textnormal{int}\gamma^{\prime}$

Here $S^{\prime}_{+}$ (resp. $S_{-}^{\prime}$) is the union of the components
of $\partial\eta(S)$ whose normal vector points out of (resp. into)
$M^{\prime}$.

We say that $(M,R_{+},R_{-},\gamma)\stackrel{{\scriptstyle
S}}{{\leadsto}}(M^{\prime},R^{\prime}_{+},R^{\prime}_{-},\gamma^{\prime})$ is
a sutured manifold decomposition, and $(M,\gamma)\stackrel{{\scriptstyle
S}}{{\leadsto}}(M^{\prime},\gamma^{\prime})$ for short.

We define a taut sutured manifold closely related to taut surfaces.

######  Definition 2.12.

A sutured manifold $(M,\gamma)$ is taut if $M$ is irreducible and $R_{+}$ and
$R_{-}$ are both taut surfaces in $M$.

######  Lemma 2.13 ([Gab87, Lemma 0.4]).

Suppose $(M,\gamma)$ is a sutured manifold and
$(M,\gamma)\stackrel{{\scriptstyle
S}}{{\leadsto}}(M^{\prime},\gamma^{\prime})$ is a sutured manifold
decomposition. If $(M^{\prime},\gamma^{\prime})$ is taut, then so is
$(M,\gamma)$.

######  Definition 2.14.

Let $(M,\gamma)$ be a sutured manifold. A _product annulus_ is an annulus $A$
in $M$ which does not cobound a solid cylinder in $M$ and such that one
boundary component of $A$ lies on $R_{-}$ and the other one lies on $R_{+}$. A
_product disk_ is a disk $D$ in $M$ such that $|D\cap s(\gamma)|=2$.

######  Definition 2.15.

Let $S\subset(M,\gamma)$ be a decomposition surface such that each component
$J$ of $S$ is a product disk or a product annulus. Then we call $S$ a _product
decomposition surface_.

Product disks and product annuli have a nice property of preserving the
tautness.

######  Lemma 2.16 ([Gab83, Lemma 3.12]).

Let $(M,\gamma)\stackrel{{\scriptstyle
J}}{{\leadsto}}(M^{\prime},\gamma^{\prime})$ be a decomposition such that
either $J$ is a product disc or a product annulus. Then $(M,\gamma)$ is taut
if and only if $(M^{\prime},\gamma^{\prime})$ is taut.

Gabai [Gab83, Definition 4.11] associates to each connected sutured manifold
$(M,\gamma)$ a complexity $c(M,\gamma)$ which is given by the 4-tuple
$(c_{1},c_{2},c_{3},c_{4})$($c_{i}$ may be denoted $c_{i}(M,\gamma)$) where
$c_{i}$ is a nonnegative integer, $c_{2}$ is a 6-tuple of nonnegative integers
with the dictionary ordering, and $c_{3}$ and $c_{4}$ are finite, possibly
empty, sets of positive integers. The set of complexities is given by the
dictionary ordering.

######  Definition 2.17.

Let $(M,\gamma)$ be a taut sutured manifold, and let $S$ be a maximal set of
pairwise disjoint non-parallel product annuli and product disks in $M$. Let
$(\overline{M},\overline{\gamma})$ be the sutured manifold obtained from
$(M,\gamma)$ by decomposing along $S$ and throwing away product sutured
manifold components which we call _windows_. Then
$(\overline{M},\overline{\gamma})$ is called the guts of $(M,\gamma)$ (which
Gabai call the reduced sutured manifold of $(M,\gamma)$). The manifold
$(\overline{M},\overline{\gamma})$ is well-defined up to isotopy by JSJ
decomposition ([JS79],[Joh06, Chapter III]).

Define the reduced complexity $\overline{C}(M,\gamma)$ of the taut manifold
$(M,\gamma)$ by $\overline{C}(M,\gamma)=c(\overline{M},\overline{\gamma})$.

In this paper, we use reduced complexity rather than complexity of a sutured
manifold and we briefly call reduced complexity as complexity.

Friedl and Kitayama [FK14] cited the following theorem in [Gab83, Section 4]
which says the complexity of a taut sutured manifold will decrease or remain
the same after decomposition.

######  Theorem 2.18.

Let $(M,\gamma)$ be a connected sutured manifold and let
$(M,\gamma)\overset{S}{\rightsquigarrow}(M^{\prime},\gamma^{\prime})$ be a
sutured manifold decomposition along a connected decomposition surface $S$.
Suppose that $(M,\gamma)$ and $(M^{\prime},\gamma^{\prime})$ are taut. Let
$(M^{\prime}_{0},\gamma^{\prime}_{0})$ be a component of
$(M^{\prime},\gamma^{\prime})$. Then

$\overline{C}(M^{\prime}_{0},\gamma^{\prime}_{0})\leq\overline{C}(M,\gamma).$

Furthermore, if $S$ is not boundary parallel, e.g. if $[S]$ is non-trivial in
$H_{2}(M,\partial M;\mathbb{Z})$, then

$\overline{C}(M^{\prime}_{0},\gamma^{\prime}_{0})<\overline{C}(M,\gamma).$

Among norm-minimizing and taut surfaces, we are more interested in classes
with nice properties called properly norm-minimizing and properly taut. This
is equivalent to (2) and (3) in Definition 2.11 for decomposition surfaces.

######  Definition 2.19.

(Cf. [Mil19]) Let $M$ be a compact 3-manifold whose boundary is a collection
of tori. Let $S$ be an oriented surface in $M$. We say that $S$ is _properly
norm-minimizing_ if $S$ is norm-minimizing and on each boundary component $P$
of $M$, all components of $\partial S\cap P$ have the same orientation
(parallel on $P$).

######  Definition 2.20.

Let $(M,R_{+},R_{-},\gamma)$ be a sutured manifold. We say that $(S,\partial
S)\subset(M,\gamma)$ is _properly taut_ if $S$ is taut and on each component
$P$ of $\gamma$, all components of $\partial S\cap P$ have the same
orientation, and moreover, the orientation is the orientation of the core of
$P$ if $P$ is an annulus.

######  Remark 2.21.

When $M$ is a manifold with toral boundary, we think of $\gamma$ as $\partial
M$ and $R_{\pm}$ is empty.

##  3\. Guts Components as Sutured Manifolds

In the following, we denote $M$ as a compact, orientable, connected,
irreducible 3-manifold with toral boundary. Furthermore, we require $x_{M}$ to
be non-degenerate. Consider $(M,\partial M)$ as a sutured manifold with
$\gamma=\partial M$.

######  Definition 3.1.

Suppose $z$ is a primitive element in $H_{2}(M,\partial M;\mathbb{Z})$. Take a
maximal collection of disjoint, non-parallel, properly norm-minimizing
decomposition surfaces $\Sigma_{1},\ldots,\Sigma_{k}$ in $M$ such that

$[\Sigma_{i}]=z\textnormal{ in }H_{2}(M,\partial M;\mathbb{Z}),\ \forall
i=1,\ldots,k.$

We call the union of $\Sigma_{1},\ldots,\Sigma_{k}$ a _facet surface_ for $z$
and denote $\Sigma_{1}\cup\ldots\cup\Sigma_{k}$ as $F(z)$.

######  Definition 3.2.

Let $S$ be a properly norm-minimizing surface in $M$ and consider
$M\backslash\backslash S$ as a sutured manifold. We define the _guts_ of $S$
as the guts of $M\backslash\backslash S$ in Definition 2.17, denoted by
$\Gamma(M,S)$.

######  Definition 3.3.

_Equivalence of guts._ Two guts $G_{1}$ and $G_{2}$ are equivalent if there
exists an isotopy of $M$ which restricts to a homeomorphism $G_{1}\to G_{2}$
as sutured manifolds.

We are going to show some properties of guts.

######  Definition 3.4.

A properly embedded surface $S$ in a sutured manifold $(M,\gamma)$ is called a
horizontal surface if i) $\partial S\subset\gamma$ and $\partial S$ is
isotopic to $\partial R_{+}(\gamma)$, ii) $S$ is taut, iii)
$[S]=[R_{+}(\gamma)]$ in $H_{2}(M,\gamma)$.

######  Definition 3.5 ([Juh08a, Definition 9.3]).

A sutured manifold is called _reduced_ if it doesn’t contain any essential
product annulus or product disk, and _horizontally prime_ if it doesn’t
contain any horizontal surface which is not parallel to boundary.

######  Lemma 3.6.

For a facet surface $F$, each component of $\Gamma(M,F)$ is reduced and
horizontally prime.

###### Proof.

First, the reason that each component $(G,\gamma)$ of $\Gamma(M,F)$ is reduced
is because we decompose along a maximal collection of product decomposition
surfaces. Next, suppose that there exists a taut surface $\Sigma$ in $G$ which
is not parallel to the boundary of $G$ but homologous to $R_{+}$ and $R_{-}$.
Take a union $F^{\prime}$ of some components of $F$ such that $F^{\prime}$
contains the $R_{+}$ part of $(G,\gamma)$ and represents a multiple of $[F]$
in $H_{2}(M,\partial M)$. We cut $R_{+}$ out of $F^{\prime}$ and glue with
$\Sigma$ to have a new surface $F^{\prime\prime}$. Then $F^{\prime\prime}$ is
homologous to a multiple of $F$, disjoint with $F$ and is non-parallel to $F$
which violates the maximality of $F$.

∎

######  Lemma 3.7.

Let $F$ be a facet surface in $M$. Then every connected component of
$M\backslash\backslash F$ contains at most one guts component.

###### Proof.

Suppose a component of $M\backslash\backslash F$ contains at least 2 guts
components. Denote the component as $N$ and consider it as a sutured manifold
$(N,R_{+}(N),R_{-}(N),\gamma(N))$. Let $(G,R_{+}(G),R_{-}(G),\gamma(G))$ be a
guts component in $(N,R_{+}(N),R_{-}(N),\gamma(N))$. Then $R_{\pm}(G)$ is a
subset of $R_{\pm}(N)$.

Take a union $F^{\prime}$ of some components of $F$ such that $F^{\prime}$
contains the $R_{+}(N)$ part of $N$ and represents a multiple of $[F]$ in
$H_{2}(M,\partial M)$. We cut $R_{+}(G)$ out of $F^{\prime}$ and glue with
$R_{-}(G)$ to have a new surface $F^{\prime\prime}$. Then $F^{\prime\prime}$
is homologous to a multiple of $F$, disjoint with $F$ and is non-parallel to
$F$ which violates the maximality of $F$. See Figure 1.

Figure 1.

∎

For the guts associated to facet surfaces representing an element in
$H_{2}(M,\partial M)$, we have the following theorem.

######  Theorem 1.1.

Let $M$ be an irreducible, orientable $3$-manifold with boundary a disjoint
union of tori $\sqcup_{i=1}^{n}P_{i}$ and non-degenerate Thurston norm. The
guts $\Gamma(M,F)$ for facet surfaces representing a primitive element $z$ in
$H_{2}(M,\partial M;\mathbb{Z})$ do not depend on the selection of facet
surfaces and product decomposition surfaces up to the equivalence of guts.

Before proving the theorem, we need some prerequisites.

######  Definition 3.8.

Let $(M,R_{+},R_{-})$ be a sutured manifold. Let $A$ be an essential annulus
in $M$ with boundary on $R_{+}\cup R_{-}$. We say $A$ is a _non-product
annulus_ if it is neither a product annulus nor boundary-parallel to
$R_{+}\cup R_{-}$.

######  Remark 3.9.

Let $S$ be a facet surface and $A$ be a non-product annulus in
$M\backslash\backslash S$. Let $N$ be the component of $M\backslash\backslash
S$ which contains $A$. We can do a surgery along $A$ for $S$ to have
$S^{\prime}$ as in Figure 2A.

A A schematic graph of a non-product annulus.

B A schematic graph of 2 non-parallel non-product annuli

Figure 2.

There are two possibilities:

  1. (1)

$S^{\prime}$ has no null-homologous component. In this case, the guts of $N$
is a solid torus with 4 longitudinal sutures which we call a _4-ST_ in this
paper,

  2. (2)

$S^{\prime}$ has null-homologous components. In this case, $A$ is null-
homologous rel boundary in $N$. Furthermore, the guts of $N$ is a sutured
manifold with toral boundary, only two sutures on a boundary component and the
other boundary components as sutured tori.

######  Lemma 3.10.

Let $S$ be a facet surface. Then non-product annuli in a component $N$ of
$M\backslash\backslash S$ are parallel to each other up to orientation.

###### Proof.

Suppose there are two non-product annuli $A$ and $A^{\prime}$ which are non-
parallel. See Figure 2B. We do a surgery along $A$ for $S$ and drop the
(possibly) null-homologous components to have $S^{\prime}$. Do the operation
again along $A^{\prime}$ for $S^{\prime}$ to have $S^{\prime\prime}$. Then by
the previous remark, between $S$ and $S^{\prime}$ there is a guts component
and so is between $S^{\prime}$ and $S^{\prime\prime}$. This violates Lemma
3.7. ∎

######  Definition 3.11.

Let $S$ and $T$ be two facet surface representing a primitive class $z$ in
$H_{2}(M,\partial M)$. Furthermore, $S$ and $T$ are in minimal position. We
call a subsurface $Q\subset S\cup T$ from decomposing $S\cup T$ along $S\cap
T$ a _patch_. For a boundary component $P$ of a component $J$ of
$M\backslash\backslash(S\cup T)$, if the co-orientation of $P\cap S$ is the
same as the co-orientation of $P\cap T$, we call $P$ a _bad boundary_ and $J$
a _bad region_. See Figure 3A as an example.

A An example of a bad region and a bad boundary.

B

Figure 3.

######  Lemma 3.12.

Under the same hypotheses as Definition 3.11, for a bad boundary $P$, $P$ is a
torus and it must satisfy one of the following conditions:

  1. (1)

it consists of only two patches, or

  2. (2)

it consists of 4 patches and the bad region $J$ whose boundary contains $P$ is
a 4-ST.

###### Proof.

By the definition of the double curve sum, $P$ is a subset of $S\oplus T$.
From Lemma 2.8, $S\oplus T$ minimizes the Thurston norm. Since $\partial J$ is
null-homologous, $P$ is a torus.

Suppose $P$ contains 4 or more patches. We denote the patches on $P$ as

$U_{1},V_{1},U_{2},V_{2},\ldots,U_{k},V_{k}$

in order where $U_{i}$ is on $P\cap S$ and $V_{i}$ is on $P\cap T$. Then each
$V_{i}$ is a non-product annulus in $M\backslash\backslash S$ and by Lemma
3.10, $V_{1},V_{2},\ldots,V_{k}$ are parallel to each other. In the
neighborhood $I\times A$ (where $A$ is an annulus) of
$V_{1},V_{2},\ldots,V_{k}$, each $V_{i}$ is homeomorphic to a copy of $A$.
Since $J$ is connected, it can only contain two $V_{i}$ and hence $J$ is a
solid torus with 4 longitudinal patches on $\partial J$.

∎

We actually need a slight generalization of the previous lemma.

######  Lemma 3.13.

Under the same hypotheses as Definition 3.11, let $N$ be a component of
$M\backslash\backslash S$ and $P_{1},\ldots,P_{k}$ are the bad boundaries with
respect to $S$ and $T$ in $N$. Then there are $k$ regions of
$N\backslash\backslash T$ such that the guts of each is a 4-ST which can be
isotoped to a neighborhood of $P_{i}$ and furthermore, its suture is isotoped
to a patch of $P_{i}$.

###### Proof.

We already prove the case when $k=1$. So we assume $k\geq 2$.

By Lemma 3.10, we know that the components of $P_{i}\cap T$ are parallel to
each other up to orientation. Furthermore, since there is at most one bad
boundary which consists of only two patches, we can assume
$P_{1},\ldots,P_{k}$ are already in order as in each case of Figure 4.
Furthermore $P_{1}\cap T,P_{2}\cap T,\ldots,P_{k}\cap T$ are contained in a
solid torus whose boundary is the union of 4 annuli $U_{1},V_{1},U_{2},V_{2}$
where $U_{i}$ is a subset of $S$, $V_{1}$ is a patch of $P_{1}\cap T$ and
$V_{2}$ is a patch of $P_{k}\cap T$. Then for each $i$, between $P_{i}$ and
$P_{i+1}$, there exists a 4-ST.

A First case.

B Second case.

Figure 4. $S$ is in blue and the non-product annuli components of $T\cap N$
are in red.

Let $S_{0}$ be the union of components of $S$ which intersect $V_{1}\cup
V_{2}$. We remove $U_{1}$ and $U_{2}$ from $S_{0}$ and attach $V_{1}$ and
$V_{2}$ to have a homologous surface $S^{\prime}_{0}$. Let $S^{\prime}_{i}$
denote the component of $S^{\prime}_{0}$ which contains $V_{i}$. Since $S_{0}$
is homologically nontrivial, at least one of $S^{\prime}_{1}$ and
$S^{\prime}_{2}$ is also homologically nontrivial. Without loss of generality,
we assume $S^{\prime}_{2}$ is homologically nontrivial. Then by the maximality
of $S$, $S^{\prime}_{2}$ is isotopic to a component $S_{1}$ of $S$.
Furthermore, $P_{k}$ can not consist of only two patches because
$S^{\prime}_{2}$ is not a torus. Thus by Lemma 3.12, $P_{k}$ consists of 4
patches and it is the boundary of a 4-ST. Then the region $J$ bounded by
$V_{2}$ and $S_{1}$ has a 4-ST guts. Even if there are components of $T\cap N$
inside $J$, they do not intersect the guts. Hence we can find a region of
$N\backslash\backslash T$ inside $J$ such that its guts is a 4-ST.

Since we can assign the 4-ST between $P_{i}$ and $P_{i+1}$ to $P_{i}$, we find
$k$ distinct regions that satisfy the condition.

∎

######  Definition 3.14.

Under the same hypotheses as Definition 3.11, let $\Gamma$ be the collection
of the guts components of $M\backslash\backslash(S\cup T)$. By Lemma 3.13, for
each bad region, there is a 4-ST guts component of $\Gamma$ in the same
component of $M\backslash\backslash S$. We remove the bad regions and their
corresponding 4-ST guts components from $\Gamma$. Then we call the remaining
guts components the _belly_ of $S$ and $T$ and denote it as
$\mathcal{B}(S,T)$.

######  Lemma 3.15.

Let $S$ and $T$ be two facet surface representing a primitive class $z$ in
$H_{2}(M,\partial M)$. Furthermore, $S$ and $T$ are in minimal position. Then
$\mathcal{B}(S,T)$ is equivalent to $\mathcal{B}(T,S)$ as sutured manifolds up
to isotopy.

###### Proof.

The only difference between $\mathcal{B}(S,T)$ and $\mathcal{B}(T,S)$ is that
we remove different equivalent 4-ST’s related to the bad regions. That does
not affect the equivalence of $\mathcal{B}(S,T)$ and $\mathcal{B}(T,S)$. ∎

######  Remark 3.16.

In the definition of the belly $\mathcal{B}(S,T)$, we remove the bad regions
and their corresponding 4-ST guts components from $\Gamma$. This is equivalent
to say that if we have two parallel non-product annuli with co-orientation
pointing each other, we can remove one of them. Hence after this operation, we
do not have two parallel non-product annuli with co-orientation pointing each
other. We further remove duplicate parallel non-product with same orientation
since they do not affect the guts of the resulting manifold from decomposing
$M\backslash\backslash S$ along the remaining patches of
$T\backslash\backslash S$. Hence for each group of parallel non-product annuli
(without orientation) of $T\backslash\backslash S$, there remain at most two
non-product annuli. If there remain two non-product annuli, their co-
orientations point outward and they bound a 4-ST. If we remove any one of
them, the guts does not change.

Therefore, we can replace each group of parallel non-product annuli (without
orientation) of $T\backslash\backslash S$ by only one non-product annulus to
have $T^{\prime}$ such that the resulting sutured manifold from decomposing
$M\backslash\backslash S$ along $T^{\prime}$ has no bad boundary and its guts
is equivalent to $\mathcal{B}(S,T)$. From now on, we use this definition for
$\mathcal{B}(S,T)$.

######  Lemma 3.17.

Let $S$ and $T$ be two facet surfaces representing a primitive class $z$ in
$H_{2}(M,\partial M)$. Furthermore, $S$ and $T$ are in minimal position. Then
the guts with respect to $S$ is equivalent to the belly $\mathcal{B}(S,T)$.

###### Proof.

By Lemma 2.8, $S\oplus T$ minimizes the Thurston norm among surfaces
representing $[S]+[T]$ (though it might contains null-homologous tori).

Let $T_{0}$ be the union of patches of $T\backslash\backslash S$ with negative
Euler characteristic and $N$ be $M\backslash\backslash S$. We want to show
that after the sutured decomposition along $T_{0}$, $N\backslash\backslash
T_{0}$ is taut.

Since $S\oplus T$ minimizes the Thurston norm, $N\backslash\backslash T_{0}$
is irreducible. We suppose a component $N_{0}$ of $N\backslash\backslash
T_{0}$ is not taut. Without loss of generality, we assume $R_{+}(N_{0})$ is
not taut. Then there is a surface $R^{\prime}$ that is homologous to
$R_{+}(N_{0})$ in $H_{2}(N_{0},\partial R_{+}(N_{0}))$ and has smaller
Thurston norm. We can do a cut-and-paste surgery on $S\oplus T$ that removes
$R_{+}(N_{0})$ and pastes back $R^{\prime}$. The resulting surface is
homologous to $S\oplus T$ but with smaller Thurston norm, which is a
contradiction.

By [Sch89, Lemma 3.2], there exists a positive integer $k$ such that we can
isotope $kS\oplus T$ away from $S$. Furthermore, as described in the proof of
the lemma, the isotopy only moves the part of $kS\oplus T$ which is inside a
tubular neighborhood $\eta(S)$ of $S$ and fix the rest. Note that $T_{0}$ is
$T\backslash\backslash S$. Then during the isotopy, $T_{0}$ is always inside
$N$.

We drop the null-homologous components of $kS\oplus T$ and denote the rest as
$T^{\prime}$. Because $T$ is homologous to $S$, $T^{\prime}$ is homologous to
a multiple of $S$. Since $S$ is a facet surface, we know that each component
of $T^{\prime}$ is boundary parallel to $N$. Here, we say a surface $U$ is
boundary parallel if there is an isotopy of $U$ onto the boundary of $N$ where
we allow moving the boundary of $U$. Since each component of $T_{0}$ has
negative Euler characteristic, $T^{\prime}$ still contains $T_{0}$ and hence
$T_{0}$ is boundary parallel in $N$ too.

Let $N_{i}$ be a component of $N$ and
$T_{0,i}\stackrel{{\scriptstyle\Delta}}{{=}}T_{0}\cap N_{i}$. We denote the
isotopy of $T_{0,i}$ onto the boundary of $N_{i}$ as $f_{t}:T_{0,i}\to
N_{i},t\in[0,1]$ where $f_{0}=\mathds{1}$ and
$T^{\prime}_{0,i}\stackrel{{\scriptstyle\Delta}}{{=}}f_{1}(T_{0,i})\subset\partial
N_{i}$. Because the isotopy of $T_{0,i}$ comes from the isotopy of
$T^{\prime}$, the images $f_{t}(T_{0,i})$ for different $t$ do not intersect
each other outside of $\partial N_{i}$. Hence the isotopy restricted on the
unfixed components of $\partial T_{0,i}$ induces a collection $A_{i}$ of
embedded annuli and $\partial A_{i}\subset\partial T_{0,i}\cup\partial
T^{\prime}_{0,i}$. See Figure 5 for a schemetic point of view. We consider two
cases of $A_{i}$:

A First case.

B Second case.

Figure 5. A shematic view of the isotopy of $T_{0,i}$. $R_{-}$ and $R_{+}$ of
$N_{i}$ are in red and blue, respectively. $T_{0,i}$ is in green and
$T^{\prime}_{0,i}$ is in purple.

  1. (1)

$A_{i}$ only consists of product annuli or boundary parallel annuli. See
Figure 5A. For each component $T_{0,i,j}$ of $T_{0,i}$, we let $A_{i,j}$ be
the union of components of $A_{i}$ induced from $\partial T_{0,i,j}$ and
$N^{\prime}_{i,j}$ be the resulting sutured manifold from decomposing $N_{i}$
along the essential components of $A_{i,j}$. In order to show that decomposing
$N_{i}$ along $T_{0,i,j}$ does not change the guts of $N_{i}$, we only need to
show that the guts of decomposing along $T_{0,i,j}\cup A_{i,j}$ is equivalent
to the guts of decomposing along $A_{i,j}$, which is equivalent to the guts of
$N_{i}$. We think of the decomposition along $T_{0,i,j}\cup A_{i,j}$ as first
decomposing $N_{i}$ along $A_{i,j}$ and then decomposing $N^{\prime}_{i,j}$
along $T_{0,i,j}\backslash\backslash A_{i,j}$. Since
$T_{0,i,j}\backslash\backslash A_{i,j}$ is boundary parallel in
$N^{\prime}_{i,j}$ with boundary on the sutured annuli of $N^{\prime}_{i,j}$,
we know that decomposing $N^{\prime}_{i,j}$ along
$T_{0,i,j}\backslash\backslash A_{i,j}$ only creates product sutured manifold
components. Thus we prove that decomposing $N_{i}$ along $T_{0,i}$ does not
change the guts of $N_{i}$.

  2. (2)

$A_{i}$ contains an essential non-product annulus $A_{0,i}$. See Figure 5B.
Then, as in Remark 3.9, we know that the guts $G_{i}$ of $N_{i}$ is a 4-ST or
a sutured manifold with toral boundary and 2 sutures on a boundary component.
Let $C_{0,i}$ be the boundary component of $A_{0,i}$ in $T_{0,i}$. Because
each component of $T_{0,i}$ either contains $C_{0,i}$ or is disjoint with it,
the algebraic intersection number between any component of $T_{0,i}$ and
$C_{0,i}$ is 0. Since each suture of $G_{i}$ is isotopic to $C_{0,i}$, the
algebraic intersection number between any component of $T_{0,i}$ and any
suture of $G_{i}$ is 0 too. Noting that the boundary of $G_{i}$ is a torus,
$T_{0,i}\cap\partial G_{i}$ are isotopic to the sutures of $G_{i}$. Hence we
can isotope $T_{0,i}$ so that $T_{0,i}$ only intersects $\partial G_{i}$ on
$\gamma(G_{i})$. Now we analyze the two possiblities of $G_{i}$ separately.

    1. (a)

If $G_{i}$ is a 4-ST, we know that $T_{0,i}\cap G_{i}$ consists of annuli
parallel to $\partial G_{i}$ and we can isotope $T_{0,i}\cap G_{i}$ into the
tubular neighborhood of $R_{+}(G_{i})$ and $R_{-}(G_{i})$ in $N_{i}$.

    2. (b)

On the other hand, suppose $G_{i}$ is a sutured manifold with toral boundary,
only two sutures on a boundary component and the other boundary components as
sutured tori. Since $T_{0,i}$ is null-homologous in $H_{2}(N_{i},\partial
N_{i})$, $T_{0,i}\cap G_{i}$ is also null-homologous in $H_{2}(G_{i},\partial
G_{i})$, i.e., $T_{0,i}\cap G_{i}$ is homologous to $R_{+}(G_{i})$ and
$R_{-}(G_{i})$ rel $\gamma(G_{i})$. Note that, by Lemma 3.6, $G_{i}$ is
horizontally prime. Hence $T_{0,i}\cap G_{i}$ is parallel to $R_{+}(G_{i})$ or
$R_{-}(G_{i})$ and thus can be isotoped into the tubular neighborhood of
$R_{+}(G_{i})$ and $R_{-}(G_{i})$ in $N_{i}$.

Figure 6. $T_{0,i}\cap G_{i}$ is isotoped into the tubular neighborhood of
$R_{+}(G_{i})$ and $R_{-}(G_{i})$ in $N_{i}$, which is in gray.

Therefore in each of these two cases, we can isotope $T_{0,i}\cap G_{i}$ into
the tubular neighborhood of $R_{+}(G_{i})$ and $R_{-}(G_{i})$ in $N_{i}$, as
in Figure 6. Since decomposing $N_{i}$ along $T_{0,i}$ obtains the guts from
decomposing $G_{i}$ along $T_{0,i}\cap G_{i}$, it only creates product sutured
manifolds and does not change the guts of $N_{i}$.

In conclusion, the guts of $N$ is equivalent to the guts of
$N\backslash\backslash T_{0}$.

Next, we consider annuli patches of $T\backslash\backslash S$. If an annulus
patch is a product annulus or has a boundary component on $\partial M$,
decomposing along it does not affect the guts. So we focus on non-product
annuli.

We replace each group of parallel non-product annuli of
$T\cap(M\backslash\backslash S)$ by only one non-product annuli with a
specific orientation as in Remark 3.16. Let $A_{1},\ldots,A_{k}$ be the non-
parallel non-product annuli. $A_{i}$ does not form a bad boundary with $S$.

Let $N_{i}$ be the component of $M\backslash\backslash S$ containing $A_{i}$.
Without loss of generality, we assume the boundary components of $A_{i}$ are
on $R_{-}(N_{i})$.

We consider separately for the two possible cases.

First, we suppose that the resulting surface from decomposing $R_{-}(N_{i})$
along $\partial A_{i}$ contains an annulus $A^{\prime}$. Then the union of
$A^{\prime}$ and $A_{i}$ is a torus. Since $A_{i}$ does not form a bad
boundary with $S$, $A_{i}$ is homologous to $A^{\prime}$ with orientation.
Then we can replace $A^{\prime}$ with $A_{i}$ from $R_{-}(N_{i})$ to have a
norm-minimizing surface $R^{\prime}$ homologous to $R_{-}(N_{i})$. By the
maximality of $S$, $R^{\prime}$ is parallel to a subsurface of
$R_{-}(N_{i})\cup R_{+}(N_{i})$. Since the guts of the resulting manifold from
decomposing $N_{i}$ along $A_{i}$ is equivalent to the guts of the resulting
manifold from decomposing $N_{i}$ along $R^{\prime}$, decomposing along
$A_{i}$ does not change the guts of $N_{i}$.

Second, we suppose each component of the resulting surface from decomposing
$R_{-}(N_{i})$ along $\partial A_{i}$ has negative Euler characteristic. We do
a surgery for $R_{-}(N_{i})$ along $A_{i}$ to have a norm-minimizing surface
$R^{\prime}$ homologous to $R_{-}(N_{i})$. Similarly, $R^{\prime}$ is parallel
to $R_{-}(N_{i})\cup R_{+}(N_{i})$. Furthermore, the components of
$R^{\prime}$ that contains a copy of $A_{i}$ are parallel to $R_{+}(N_{i})$.
Therefore the guts of $N_{i}$ contains a 4-ST and because of Lemma 3.7, the
guts of $N_{i}$ is actually a 4-ST. Furthermore, after decomposing $N_{i}$
along $A_{i}$, the guts still remains as a 4-ST. See Figure 7 for reference.

Figure 7. Illustration of surfaces inside $N_{i}$. $R_{-}$ is in red, $A_{i}$
is in blue and $R^{\prime}$ is in pink.

In summary, decomposing $M\backslash\backslash S$ along $A_{1}\cup\ldots\cup
A_{k}$ also does not change the guts. Since we already consider all
possibilities, decomposing $M\backslash\backslash S$ along each patch of
$T\backslash\backslash S$ does not change the guts of $M\backslash\backslash
S$. The theorem is proved.

∎

###### Proof of Theorem 1.1.

Let $S,T$ be 2 facet surfaces representing $z$ and isotope $S,T$ to be in
minimal position. By Lemma 3.17, $\Gamma(M,S)$ is equivalent to
$\mathcal{B}(S,T)$ and $\Gamma(M,T)$ is equivalent to $\mathcal{B}(T,S)$.
Hence by Lemma 3.17, $\Gamma(M,S)$ is equivalent to $\Gamma(M,T)$. ∎

######  Definition 3.18.

For an element $z$ in $H_{2}(M,\partial M;\mathbb{Z})$, we define
$(\Gamma(z),\gamma(z))$ (or just $\Gamma(z)$) to be the guts with respect to
facet surfaces representing $z$ in Theorem 1.1.

######  Definition 3.19.

Given a taut sutured manifold $(M,R_{+},R_{-},\gamma)$, we do a sutured
decomposition along a maximal collection of horizontal surfaces and then
decompose along a maximal product decomposition surfaces. By throwing away the
components which are product sutured manifolds, we call the remaining sutured
manifolds _horizontally prime guts_ of $(M,\gamma)$.

Using the same method in the proof of Theorem 1.1, we are able to show

######  Corollary 3.20.

Let $(M,\gamma)$ be a taut sutured manifold such that there is no non-
separating annulus or torus rel $\gamma$. The horizontally prime guts of
$(M,\gamma)$ is well defined up to equivalence.

##  4\. Guts and Thurston Cones

In this section, we are going to prove Theorem 1.2.

######  Definition 4.1.

Let $M$ be a $3$-manifold with boundary a disjoint union of tori
$\sqcup_{i=1}^{n}P_{i}$ and $\alpha$ be an element in $H_{2}(M,\partial
M;\mathbb{R})$. Let $\partial:H_{2}(M,\partial M;\mathbb{R})\to H_{1}(\partial
M;\mathbb{R})$ be the map from the long exact sequence of $(M,\partial M)$. We
denote the restriction of $\alpha$ on a boundary component $P_{i}$ as the
coordinate of $\partial\alpha$ in $H_{1}(P_{i};\mathbb{R})$. We say two non-
zero elements $u_{1},u_{2}\in H_{1}(P_{i};\mathbb{R})$ are in opposite
orientation if $u_{1}$ is a negative multiple of $u_{2}$.

######  Theorem 1.2.

Let $M$ be an irreducible, orientable $3$-manifold with boundary a disjoint
union of tori $\sqcup_{i=1}^{n}P_{i}$ and non-degenerate Thurston norm. Let
$y,z$ be two elements in an open face of the Thurston sphere. If there is an
open segment $(v,w)$ containing $y,z$ in the open face such that the
restrictions of $v$ and $w$ on each boundary component are not in opposite
orientations, the guts $\Gamma(y)$ is equivalent to $\Gamma(z)$.

Before proving the theorem, we need a slight modification of [Mil19,
Proposition 2.8] and a lemma.

######  Proposition 4.2.

Let $\alpha_{1}$ and $\alpha_{2}$ be in the same closed Thurston face of a
non-degenerate Thurston norm $x_{M}$, where $M$ is a $3$-manifold with
boundary a disjoint union of tori $\sqcup_{i=1}^{n}P_{i}$. If the restriction
of $\alpha_{1}$ and $\alpha_{2}$ on each boundary component $P_{j}$ are not in
opposite orientation, then there exist properly norm-minimizing surfaces
$S_{1}$ and $S_{2}$ with $[S_{i}]=\alpha_{i}$ so that for any positive
integers $a$ and $b$, $aS_{1}\oplus bS_{2}$ is properly norm-minimizing.

###### Proof.

(Cf. the proof of [Mil19, Proposition 2.8]) Since $x_{M}$ is nondegenerate,
$\chi_{-}(R)=-\chi(R)$ for any norm-minimizing surface $R$ in $M$.

Let $S_{1}$ and $S_{2}$ be properly norm-minimizing surfaces with
$[S_{i}]=\alpha_{i}$. Isotope the $S_{i}$ near their boundaries so that
$\partial S_{1}$ and $\partial S_{2}$ intersect minimally. Because the
restriction of $\alpha_{1}$ on each boundary component $P_{i}$ is not a
negative multiple of the restriction of $\alpha_{2}$, we know that for each
$j$, every component of $\partial(aS_{1}\oplus bS_{2})\cap P_{j}$ is
nontrivial and have the same orientation which means it represents the same
homology class. We have
$x_{M}(a[S_{1}]+b[S_{2}])=ax_{M}([S_{1}])+bx_{M}([S_{2}])=a\chi_{-}(S_{1})+b\chi_{-}(S_{2})=-a\chi(S_{1})-b\chi(S_{2})=-\chi(aS_{1}\oplus
bS_{2})$. Then we are done if $aS_{1}\oplus bS_{2}$ has no disk, $2$-sphere,
torus or annulus components.

Suppose $S_{1}\setminus S_{2}$ includes a component $C$ which does not meet
$\partial S_{1}$ and with $\chi(C)\geq 0$. Then surger $S_{2}$ along $C$ and
throw away null-homologous components to obtain $S^{\prime}_{2}$ (i.e.
$S^{\prime}_{2}:=[S_{2}\setminus((\partial C)\times I)]\cup(C\times S^{0})$).
The surface $S^{\prime}_{2}$ is homologous to $S_{2}$. Since $\chi(C)\geq 0$,
$S^{\prime}_{2}$ is norm-minimizing. Set $S_{2}:=S^{\prime}_{2}$ and repeat
until $S_{1}\setminus S_{2}$ includes no such component $C$. Now any closed
components of $aS_{1}\oplus bS_{2}$ must include a region homeomorphic to some
component of $S_{1}\setminus S_{2}$, which must have negative Euler
characteristic. Therefore, $aS_{1}\oplus bS_{2}$ has no closed sphere or torus
components.

Since every component of $\partial(aS_{1}\oplus bS_{2})\cap P_{j}$ is
nontrivial, $aS_{1}\oplus bS_{2}$ has no disk components. Suppose
$aS_{1}\oplus bS_{2}$ has an annulus component $A$, by the non-degeneracy of
$x_{M}$, $A$ is homologically trivial in $H_{2}(M,\partial M)$ which means
$\partial A$ is on a boundary component $P_{j}$ and is homologically trivial
in $H_{1}(P_{j})$ which violates the fact that the two components of $\partial
A$ have the same orientation.

∎

We state a simple lemma without proof.

######  Lemma 4.3.

Let $(N,R_{+},R_{-},\gamma)$ be a product sutured manifold with $\chi(N)<0$.
If $S$ is a properly embedded surface without annulus components such that $S$
is homologous to $R_{+}$ in $H_{2}(N,\gamma)$ and $\chi_{-}(S)=\chi_{-}(R+)$,
then $S$ is parallel to $R_{+}$.

###### Proof of Theorem 1.2.

Let $y$ and $z$ be in the same open Thurston cone and $(v,w)$ be an open
segment containing the closed segment $[y,z]$ in the same cone such that the
restrictions of $v$ and $w$ on each boundary component are not in opposite
orientations. By Proposition 4.2, we can take properly norm-minimizing
representatives $S_{v}$ and $S_{w}$ for $v$ and $w$ respectively such that
$S_{v}\backslash S_{w}$ and $S_{w}\backslash S_{v}$ has no component of
nonnegative Euler characteristic. Moreover, for any positive integers $a$ and
$b$, $aS_{v}\oplus bS_{w}$ is properly norm-minimizing.

Similar to [Oer86, Proposition 8], we can create a properly taut oriented
branched surface $B$ carrying $S_{v}$ and $S_{w}$ by changing each curve of
$S_{v}\cap S_{w}$ to an “annulus or disk of contact”. $B$ also carries
$aS_{v}\oplus bS_{w}$ for any positive integers $a$ and $b$. Let $N_{B}$ be
the fibered neighborhood of $B$ and we decompose $\partial N_{B}$ into two
parts: horizontal boundary $\partial_{h}N_{B}$ which is transversal to the
fibers and vertical boundary $\partial_{v}N_{B}$ which consists of some
fibers. See Figure 8.

Figure 8. Schematic figure for the branched surface $B$.

Let $S$ be the surface $aS_{v}\oplus bS_{w}$ for some $a$ and $b$. As in
[TW96, Section 6], we isotope $2S$ so that it contains
$int_{2S}(\partial_{h}N_{B})$ where $int_{2S}(\partial_{h}N_{B})$ is the
interior of $\partial_{h}N_{B}$ in $2S$ and let $L_{S}$ denote the $I$-bundle
obtained by splitting $N_{B}$ along $2S-int_{2S}(\partial_{h}N_{B})$. Then the
vertical boundary of $L_{S}$ is the union of $\partial M$ and
$\partial_{v}N_{B}$.

We want to show no component of $L_{S}$ is $K\times I$, where $K$ is a
punctured disk in the interior of $2S$ and there is a boundary curve $\alpha$
of $K$ such that $\alpha\times\\{0\\}$, $\alpha\times\\{1\\}$ bound disks
$D_{0},D_{1}$ in $2S$ containing $K\times\\{0\\},K\times\\{1\\}$,
respectively. Otherwise, suppose there is such a component $K\times I$. Since
$K$ is in the interior of $2S$, $\partial D_{0}$ is in $\partial_{v}N_{B}$.
Because $\partial_{v}N_{B}$ is isotopic to $(S_{u}\cap S_{w})\times I$, we can
isotope $D_{0}$ such that $\partial D_{0}$ is a component of $S_{v}\cap
S_{w}$. Since $S_{v}$ is incompressible, $\partial D_{0}$ bounds a disk in
$S_{v}$. This means there is a disk component in $S_{v}\backslash S_{w}$,
which is a contradiction to the construction of $S_{v}$ in Proposition 4.2.

We can also obtain $L_{S}$ as follows. First, we decompose $M$ along $2S$ and
then further decompose along $\partial_{v}N_{B}$. The resulting manifold is
the union of $L_{S}$ and $M\backslash N_{B}$. Let $A$ be an annulus component
of $\partial_{v}N_{B}$. Then by the preceding paragraph $\partial A$ is
nontrivial in $2S$ and hence $A$ is a product annulus. Therefore
$\partial_{v}N_{B}$ is a union of some product annuli and product disks.

Let $F$ be a facet surface that represents $av+bw$ and contains $S$. Isotope
$2F$ such that it intersects minimally the product annuli and product disks
that separate $L_{S}$ and $M\backslash N_{B}$. For each component $J$ of
$L_{S}$, we think of it as a product sutured manifold
$(J,R_{+},R_{-},\gamma)$. We want to show that, $2F$ intersects each component
$\gamma_{i}$ of $\gamma$ in the same orientation. Since $2F$ is properly norm-
minimizing, $2F\cap J$ is a taut surface homologous to a multiple of
$[R_{+}]$. By Lemma 4.3, $2F\cap J$ is parallel to $R_{+}$ and hence $2F$
intersects $\gamma$ in the same orientation.

We want to show that $2F\cap(M\backslash N_{B})$ is a maximal collection of
horizontal surfaces (with multiplicities) for $M\backslash N_{B}$. Since $2F$
intersects each of $\partial_{v}N_{B}$ and $\partial M$ in the same direction,
$2F\cap(M\backslash N_{B})$ is a union of horizontal surfaces for $M\backslash
N_{B}$. If there is a horizontal surface $F_{0}$ in a component
$(J,R_{+},R_{-},\gamma)$ of $M\backslash N_{B}$ which is disjoint with and not
parallel to $2F\cap(M\backslash N_{B})$, we can remove $R_{+}$ from $2F$ and
attach $F_{0}$ to the rest to have a surface which is disjoint with and not
parallel to $2F$. This violates the maximality of $F$.

Hence the guts of $av+bw$ is equivalent to the horizontally prime guts of
$M\backslash N_{B}$. By Corollary 3.20, the guts of $av+bw$ remain the same
for different choice of $a$ and $b$. Since we can choose $a$ and $b$ such that
$av+bw$ is a positive multiple of $y$ and $z$ respectively, $\Gamma(y)$ is
equivalent to $\Gamma(z)$.

∎

For the case when $M$ is closed, any open segment $(v,w)$ containing $y,z$ in
the open face satisfies the condition that the restrictions of $v$ and $w$ on
each boundary component are not in opposite orientations. Hence we have the
following corollary.

######  Corollary 4.4.

When $M$ is closed, let $y,z$ be two elements in an open face of the Thurston
sphere. Then the guts $\Gamma(y)$ are equivalent to $\Gamma(z)$.

##  5\. Restriction Map to the Guts in Homology

The following theorem is a generalization of what was used in Agol’s paper
about virtual fibering conjecture [Ago08, Lemma 4.1].

######  Theorem 1.3.

Let $z$ lie on a $k$-codimensional open Thurston cone $\Delta$. We denote the
subspace spanned by $\Delta$ as $V$. Then the kernel of the restriction map
$\varphi:H_{2}(M,\partial M;\mathbb{Z})\to
H_{2}(\Gamma(z),\partial\Gamma(z);\mathbb{Z})$ is a subspace of $V$.

If there is a neighborhood of $z$ in $\Delta$ which does not contain two
elements whose restrictions on a boundary component are in opposite
orientations, the kernel of $\varphi$ is exactly $V$ and the rank of the image
of $\varphi$ is $k$, i.e. the rank of the image of

$H^{1}(M)\to H^{1}(\Gamma(z))$

is $k$.

Before proving the theorem, we need several useful lemmas:

######  Lemma 5.1.

Let $z$ be an element in $H_{2}(M,\partial M;\mathbb{Z})$ and $\varphi$ be the
restriction map $\varphi:H_{2}(M,\partial M;\mathbb{Z})\to
H_{2}(\Gamma(z),\partial\Gamma(z);\mathbb{Z})$. The image of $u$ under the map
$\varphi$ is 0 if and only if we can find a surface $\Sigma$ representing a
multiple of $u$ in $H_{2}(M,\partial M;\mathbb{Z})$ avoids the guts of $z$.

###### Proof.

First, we assume $\partial M=\emptyset$. Looking at the long exact sequence
for $(M,M^{\prime})$ where
$M^{\prime}\stackrel{{\scriptstyle\Delta}}{{=}}M\backslash\mathrm{int}{\Gamma(z)}$:

$\cdots\to H_{2}(M^{\prime})\stackrel{{\scriptstyle
i_{*}}}{{\to}}H_{2}(M){\to}H_{2}(M,M^{\prime})\to H_{1}(M^{\prime})\to\cdots$

we have a short exact sequence

$H_{2}(M^{\prime})\stackrel{{\scriptstyle
i_{*}}}{{\to}}H_{2}(M)\stackrel{{\scriptstyle\varphi}}{{\to}}H_{2}(\Gamma(z),\partial\Gamma(z))$

because of the excision lemma $H_{2}(M,M^{\prime})\cong
H_{2}(\Gamma(z),\partial\Gamma(z))$.

Hence $\varphi(u)=0$ if and only if there exists a surface $\Sigma$ in
$M^{\prime}$ such that $i_{*}([\Sigma])=u$ and hence
$\varphi(i_{*}([\Sigma]))=0$, which means $\Sigma$ avoids the guts of $z$.

When $\partial M\neq\emptyset$, we have a long exact sequence

$\cdots\to
H^{1}(M,\Gamma(z)){\to}H^{1}(M)\stackrel{{\scriptstyle\varphi}}{{\to}}H^{1}(\Gamma(z))\to\cdots$

and the excision lemma

$H^{1}(M,\Gamma(z))\cong
H^{1}(M\backslash\mathrm{int}{\Gamma(z)},\partial\Gamma(z)\backslash\partial
M).$

Hence we have the following short exact sequence:

$H^{1}(M\backslash\mathrm{int}{\Gamma(z)},\partial\Gamma(z)\backslash\partial
M){\to}H^{1}(M)\stackrel{{\scriptstyle\varphi}}{{\to}}H^{1}(\Gamma(z))$

Noting that
$\partial(M\backslash\mathrm{int}{\Gamma(z)})=(\partial\Gamma(z)\backslash\partial
M)\cup(\partial M\backslash\partial\Gamma(z))$, by Poincare duality for
manifolds with boundary, we have
$H^{1}(M\backslash\mathrm{int}{\Gamma(z)},\partial\Gamma(z)\backslash\partial
M)\cong H_{2}(M\backslash\mathrm{int}{\Gamma(z)},\partial
M\backslash\partial\Gamma(z))$. Also by Poincare duality, we have
$H^{1}(M)\cong H_{2}(M,\partial M)$ and $H^{1}(\Gamma(z))\cong
H_{2}(\Gamma(z),\partial\Gamma(z))$.

Hence, we have the following exact sequence

$H_{2}(M\backslash\mathrm{int}{\Gamma(z)},\partial
M\backslash\partial\Gamma(z))\stackrel{{\scriptstyle
i_{*}}}{{\to}}H_{2}(M,\partial
M)\stackrel{{\scriptstyle\varphi}}{{\to}}H_{2}(\Gamma(z),\partial\Gamma(z))$

which plays the same role in the case when $\partial M$ is empty. ∎

######  Lemma 5.2 ([Sch89, Lemma 3.2]).

Suppose $(T,\partial T)$ and $(S,\partial S)$ are properly embedded surfaces
in general position in the 3-manifold $(M,\partial M)$, and $T$ is homologous
rel $\partial$ to a surface in $M$, which is disjoint from the interior of
$S$. Then:

(a) $\Lambda=T\cap S$ has trivial fundamental class in $H_{1}(S,\partial S)$,

(b) $T\oplus kS$ is properly isotopic to a surface disjoint from $S$ for some
sufficiently large positive integer.

######  Lemma 5.3.

Let $z$ be an element in $H_{2}(M,\partial M;\mathbb{Z})$ and $\varphi$ be the
restriction map $\varphi:H_{2}(M,\partial M;\mathbb{Z})\to
H_{2}(\Gamma(z),\partial\Gamma(z);\mathbb{Z})$. Suppose the image of $u$ under
the map $\varphi$ is 0. We can find a properly norm-minimizing surface
$\Sigma$ representing $nz+u$ in $H_{2}(M,\partial M;\mathbb{Z})$ which avoids
the guts of $z$, where $n$ is a sufficiently large positive integer.

###### Proof.

First, we find an integer $m$ such that $mz+u$ and $z$ lies in the same closed
face of Thurston’s norm unit ball. Denote $F(z)$ as a facet surface of $z$ and
$\Sigma^{\prime}$ is a properly norm-minimizing surface of $mz+u$ in $M$. By
Lemma 5.1, $\Sigma^{\prime}$ is homologous to a surface in $M^{\prime}$ where
$M^{\prime}=M\backslash\mathrm{int}{\Gamma(z)}$. Hence by Lemma 5.2, the
double curve sum $\Sigma^{\prime\prime}$ of finitely many copies of $F(z)$ and
$\Sigma^{\prime}$ in the neighborhood of $F(z)$ after an isotopy will be
disjoint with $F(z)$. Let $\gamma(z)$ be the sutures of $\Gamma(z)$. Because
$\Sigma^{\prime\prime}$ is disjoint with $F(z)$, $\Sigma^{\prime\prime}$
intersects $\gamma(z)$ in a union of essential closed curves. Note that
$\Sigma^{\prime\prime}\cap\gamma(z)$ might have components of opposite
orientation to the orientation of $\gamma(z)$. However, we can add more copies
of $F(z)$ to $\Sigma^{\prime\prime}$ and for closed curves of different
orientation in $\gamma(z)$, we can do a cut-and-paste surgery along the annuli
formed by those curves to have a new surface $\Sigma$ such that all closed
curves of $\Sigma\cap\gamma(z)$ are in the same orientation of $\gamma(z)$.

Then $\varphi([\Sigma])$ is exactly represented by the intersection
$\Sigma\cap\Gamma(z)$. We are going to show that $\Sigma\cap\Gamma(z)$ is
parallel to some components of $R_{+}(z)$ and $R_{-}(z)$. Let
$(\Gamma^{i}(z),R^{i}_{+}(z),R^{i}_{-}(z),\gamma^{i}(z))$ be the connected
components of $\Gamma(z)$ and $\Sigma^{i}=\Sigma\cap\Gamma^{i}(z)$.

By applying the long exact sequence of
$(\Gamma^{i}(z),\partial\Gamma^{i}(z),\gamma^{i}(z))$, we have

$H_{2}(\partial\Gamma^{i}(z),\gamma^{i}(z))\to
H_{2}(\Gamma^{i}(z),\gamma^{i}(z))\to
H_{2}(\Gamma^{i}(z),\partial\Gamma^{i}(z)).$

Since $[\Sigma^{i}]$ is zero in $H_{2}(\Gamma(z),\partial\Gamma(z))$, we can
find some components $R^{i}$ (with multiplicities) of $R^{i}_{+}(z)$ and
$R^{i}_{-}(z)$ which is homologous to $\Sigma^{i}$. Because $\Sigma$ is a
properly norm-minimizing surface in $M$, $\Sigma^{i}$ is taut in
$H_{2}(\Gamma^{i}(z),\gamma^{i}(z))$. Since $R^{i}_{+}(z)$ and $R^{i}_{-}(z)$
are also taut surfaces in $H_{2}(\Gamma^{i}(z),\gamma^{i}(z))$, we have
$\chi_{-}(\Sigma^{i})=\chi_{-}(R^{i})$. If we take large enough pieces of
$R_{+}(z)$ and $R_{-}(z)$ and replace $R^{i}$ with $\Sigma^{i}$, we obtain a
taut surface $\Omega^{i}$ representing a multiple of $[R^{i}_{-}]$ in
$H_{2}(\Gamma^{i}(z),\gamma^{i}(z))$. By Lemma 3.6, $\Gamma^{i}(z)$ is
horizontally prime. Since $\Omega^{i}$ consists of horizontal surfaces and
$\Sigma^{i}$ is a union of some components of $\Omega^{i}$, we know that
$\Sigma^{i}$ is parallel to some components of $R^{i}_{+}(z)$ and
$R^{i}_{-}(z)$ of $\Gamma^{i}(z)$.

Therefore, we can isotope $\Sigma^{i}$ off the guts $\Gamma^{i}(z)$ for each
$i$, i.e., isotope $\Sigma$ off the guts $\Gamma(z)$. ∎

######  Lemma 5.4.

Let $z$ be an element in $H_{2}(M,\partial M;\mathbb{Z})$ and $S$ is a
properly norm-minimizing surface representing a multiple of $z$. Let $A$ be an
essential annulus or an essential disk. If $F(z)$ is a facet surface
representing $z$ and containing $S$, we can isotope $A$ such that $A$ is
outside of the guts of $F(z)$.

###### Proof.

We isotope $A$ so that $A$ intersects $F(z)$ in minimal number of components.

When $A$ is a disk, since $F(z)$ is properly norm-minimizing, $F(z)$
intersects $A\cap\partial M$ in the same orientation, and hence $F(z)$
intersects $A$ in the same orientation. So $A\backslash\backslash F(z)$ is a
union of product disks which is outside of the interior of the guts of $F(z)$.
Therefore, we can isotope $A$ a little bit to avoid the guts.

When $A$ is an annulus, $A\backslash\backslash F(z)$ is a union of annuli. Let
$A_{0}$ be a component of $A\backslash\backslash F(z)$. Suppose $A_{0}$ is not
a product annulus for $M\backslash\backslash F(z)$ and both boundary curves of
$A_{0}$ lie on the $R_{+}$ part of $M\backslash\backslash F(z)$. Then we can
do a cut-and-paste along $A_{0}$ for $R_{+}$ so that we have a new properly
norm-minimizing surface $R^{\prime}$ after throwing the null-homologous
components. Since $F(z)$ is a facet surface, $R^{\prime}$ is parallel to some
components of $R_{-}$. As a subset of $R^{\prime}$, $A_{0}$ can be isotoped to
avoid the guts. ∎

We also need the following lemma: (Cf. [Gab87, Lemma 0.6] and [Ni11,
Proposition 2.12] )

######  Lemma 5.5.

Let $F$ be a compact surface. For any two collections of disjoint loops
$C_{0}$ and $C_{1}$ in $F$, if $[C_{0}]=[C_{1}]$ in $H_{1}(F,\partial F)$,
then there exists a sequence of collections of loops:

$C_{0}=\gamma_{0},\gamma_{1},\cdots,\gamma_{m}=C_{1}$

and a sequence of embedded surface with no components of disks and compatible
with the orientation of $F$

$W_{1},W_{1},\cdots,W_{m}$

such that

$\overline{\partial W_{i}\backslash(\partial
F)}=\gamma_{i}\cup(-\gamma_{i-1}).$

###### Proof.

Apply [Gab83, Lemma 3.10] to $C_{0}\cup(-C_{1})$. ∎

Now we can start proving Theorem 1.3.

######  Proof of Theorem 1.3.

We are going to prove the theorem by showing the two inclusions.

First, we show $\textnormal{ker }\varphi\subset V$, which will be proved by
showing that the Thurston norm for a small neighborhood of $z$ in
$\textnormal{ker }\varphi$ can be represented by an Euler class. Let $F(z)$ be
a facet surface of $z$, $M^{\prime}$ be
$M\backslash\mathrm{int}{\Gamma(M,F(z))}$ and $X\times I$ be the window with
respect to $F(z)$.

We can think of $M^{\prime}$ as the result of gluing two ends of
$F(z)\times[0,1]$ by two ends of $X\times[0,1]$, i.e.,

$M^{\prime}\cong F(z)\times[0,1]\sqcup
X\times[0,1]/[(f(x),1)\sim(x,0),(g(x),-1)\sim(x,1)]$

where $f,g$ are two maps from $X$ to $F(z)$.

Figure 9. $M^{\prime}=M\backslash\mathrm{int}{\Gamma(M,F(z))}$.

Since $u$ is in the kernel of $\varphi$, by lemma 5.3, we can find a properly
norm-minimizing surface $\Sigma$ in $H_{2}(M,\partial M)$ which represents a
multiple of $Nz+u$ such that $\Sigma$ does not intersects $\Gamma(M,F(z))$,
i.e., inside $M^{\prime}$. Besides, $Nz+u$ and $z$ lie in a closed Thurston
cone.

Since $M^{\prime}$ is a union of product sutured manifolds, i.e.,
$F(z)\times[0,1]$ and windows $X\times I$, it has a natural 1-codimensional
fibration $\mathscr{F}$. We fix a Riemannian metric on $M^{\prime}$ such that
the normal vector field $n(\mathscr{F})$ of $\mathscr{F}$ follows the
direction of the intervals. This normal vector field will be used in the
following proof.

Here we consider $\partial M=\emptyset$ first. We isotope $\Sigma$ such that
$\Sigma\cap X\times[0,1]=C\times[0,1]$, where $C$ is a collection of closed
curves because $\Sigma$ avoids the guts $\Gamma(M,F(z))$. In the following we
will write $F(z)$ as $F$.

Then $\Sigma_{0}\stackrel{{\scriptstyle\Delta}}{{=}}\Sigma\cap(F\times[0,1])$
has boundary $(C_{-}\times\\{0\\})\cup(C_{+}\times\\{1\\})$. If a component
$C_{i}$ of $C_{-}$ or $C_{+}$ is trivial in $F$, i.e. there is a disk $D$ in
$F$ whose boundary is $C_{i}$. Then by the incompressibility of $\Sigma$,
$C_{i}$ bounds a disk $D^{\prime}$ in $\Sigma$. We can isotope $D^{\prime}$ so
that we remove $C_{i}$ from the intersection of $\Sigma$ and
$F\times\\{0,1\\}$. Hence in the following, we assume no component of $C_{-}$
and $C_{+}$ is trivial. We are going to construct a new surface $\Omega_{0}$
in $F\times[0,1]$ such that $\partial\Omega_{0}=\partial\Sigma_{0}$ and
$[\Omega_{0}-\Sigma_{0}]=0$ in $H_{2}(F\times[0,1])$.

Since $C_{-}\times\\{0\\}\cup C_{+}\times\\{1\\}$ bounds $\Sigma_{0}$,
$[C_{-}\times\\{0\\}]-[C_{+}\times\\{1\\}]=0$ in $H_{1}(F\times[0,1])$, and
hence by projection, $[C_{-}]=[C_{+}]$ in $H_{1}(F)$.

We first isotope $\Sigma_{0}$ in $F\times I$ such that the intersection of
$\Sigma_{0}$ and $F\times[0,1/2]$ is $C_{-}\times[0,1/2]$. Then we will do the
surgery used in Gabai’s paper [Gab83, Lemma 3.11]. Let W be a subsurface of
$F$ such that $\partial W=\gamma^{\prime}\cup(-\gamma)$ where $\gamma$ is
$C_{-}$ and $\gamma^{\prime}$ is disjoint with $C_{-}$. Then the double curve
sum $\Sigma_{1}=\Sigma_{0}\oplus F\times(1/4)$ can be isotoped slightly so
that $\Sigma_{1}$ intersect $F\times(1/4)$ at $\gamma^{\prime}$. This can be
seen by performing the surgery in two steps. First do the surgery along the
curves $\gamma$, and isotope the resulting surface $P$ slightly so that $P\cap
F\times(1/4)=\gamma^{\prime}$. Finally do the surgery along the remaining
curves to get $\Sigma_{1}$. If the surgery obtains some spheres or disks, then
by the incompressibility of $F$ and $\Sigma_{0}$, there exist 2 disks
$D_{1}\in F$ and $D_{2}\in\Sigma_{0}$ such that $D_{1}\cup D_{2}$ is a sphere.
Because $F$ is the unique norm-minimizing surface in $F\times S^{1}$ up to
isotopy, we know that they bound a 3-cell. Hence we can isotope $\Sigma_{0}$
vertically to avoid popping out disks or spheres.

It now follows from Lemma 5.5 that by surgering $\Sigma_{0}$ and $m$ copies of
$F$ we can obtain a new surface (which we still denote as $\Sigma_{0}$)
satisfying the following properties:

1\. the normal vector of $\Sigma_{0}\cap F\times[0,1/2]$ always has an angle
less than or equal to $\pi/2$ with the normal vector $n(\mathscr{F})$ induced
from $[0,1]$;

2\. $\Sigma_{0}\cap F\times[1/2,1]$ will have the same boundary on each sides,
i.e. $\Sigma_{0}\cap F\times 1/2=\Sigma_{0}\cap F\times 1=C_{+}$.

Now we try to replace $\Sigma^{*}=\Sigma_{0}\cap F\times[1/2,1]$ out of
$\Sigma$ with some good-behaving surface. Denote $F\times\\{1/2\\}$ and
$F\times\\{1\\}$ as $F_{-}$ and $F_{+}$ respectively. Glue two boundaries of
$F\times[1/2,1]$ together and then we get a properly embedded surface
$\Sigma^{\prime}$ in $F\times S^{1}$. We can stabilize $\Sigma^{\prime}$ by
several pieces of $F_{-}$ and $F_{+}$ such that the homology class of
$\Sigma^{\prime}$ in $H_{2}(F\times S^{1})$ is represented by some oriented
components of $F$ and vertical tori. Hence we can construct a properly norm-
minimizing surface $\Omega^{\prime}$ by taking the double curve sum of some
oriented components of $F$ and vertical tori. Now we decompose
$\Omega^{\prime}$ along the $F_{-}=F_{+}$ to obtain a surface $\Omega^{*}$
having the same boundary as $\Sigma^{*}$. Hence, $\Omega^{*}\cup-\Sigma^{*}$
is a closed surface and $[\Omega^{*}\cup-\Sigma^{*}]=0$ in $H_{2}(F\times
S^{1})$, which is therefore 0 in $H_{2}(F\times[1/2,1],F_{-}\cup F_{+})$. From
the exact sequence

$H_{2}(F_{-}\cup F_{+})\to H_{2}(F\times[1/2,1])\to
H_{2}(F\times[1/2,1],F_{-}\cup F_{+})$

we know that we can stabilize $\Sigma^{*}$ and $\Omega^{*}$ with $F_{-}$ and
$F_{+}$ such that $[\Omega^{*}\cup-\Sigma^{*}]$ is 0 in
$H_{2}(F\times[1/2,1])$.

We replace $\Sigma^{*}$ out of $\Sigma$ with $\Omega^{*}$ to get a new surface
$\Omega$. We want to show $\chi_{-}(\Sigma^{*})=\chi_{-}(\Omega^{*})$. If this
is true, $\Sigma$ and $\Omega$ have the same homology class and the same
Thurston norm. For one direction, since $\Sigma$ is a properly norm-minimizing
surface in $M$, $\chi_{-}(\Sigma^{*})$ is smaller than or equal to
$\chi_{-}(\Omega^{*})$, otherwise $\chi_{-}(\Omega)<\chi_{-}(\Sigma)$. For the
other direction, this comes from $\Omega^{\prime}$ is a properly norm-
minimizing surface in $H_{2}(F\times S^{1})$.

In short, $\Omega$ is a properly norm-minimizing surface for some
$n^{\prime}z+u$, where $n^{\prime}$ is a positive integer. Now we are going to
show the Thurston norm of $z$ and $n^{\prime}z+u$ is represented by a fixed
Euler class independent of $u$.

By the construction of $\Omega$, we know that the angle between $\Omega$ and
$\mathscr{F}$ is always less than or equal to $\pi/2$.

Hence by [CC03, Lemma 10.5.7], we know that the 2-plane bundle $T\Omega$ is
isomorphic to $\mathscr{F}|\Omega$, which means

$\chi_{-}(\Omega)=-e(\mathscr{F})(\Omega)$

as well as

$\chi_{-}(F(z))=-e(\mathscr{F})(F(z))$

Thus we know that

$\displaystyle x(z)$ $\displaystyle=$ $\displaystyle-e(\mathscr{F})(z)$
$\displaystyle x(n^{\prime}z+u)$ $\displaystyle=$
$\displaystyle-e(\mathscr{F})(n^{\prime}z+u)$

Since $u$ is in arbitrary direction, the Thurston norm of a small neighborhood
of $z$ in $\textnormal{ker}\varphi$ can be represented by $-e(\mathscr{F})$,
which means they lie in the same open face of the Thurston’s norm unit ball.
Hence $\textnormal{ker}\varphi\subset V$.

When $\partial M\neq\emptyset$, we consider $H_{2}(F\times[-1,1],\partial
F\times[-1,1])$ and can show the same result.

Next, suppose there is a neighborhood of $z$ in $\Delta$ which does not
contain two elements whose restrictions on a boundary component are in
opposite orientations. We want to show $V\subset\textnormal{ker}\varphi$. In
order to prove this, for any direction in $V$, we take $u$ small enough such
that $z$, $z-u$ and $z+u$ lie in the same Thurston cone and the restrictions
of $z-u$ and $z+u$ on each boundary component are in the same orientation. We
can multiply $z$ and $u$ by an integer such that $z$, $z-u$ and $z+u$ are in
$H_{2}(M,\partial M;\mathbb{Z})$ and we still use $z$, $z-u$ and $z+u$ to
represent these integral elements.

We take properly norm-minimizing surfaces $S_{z+u}$ and $S_{z-u}$ for ${z+u}$
and ${z-u}$, respectively from Proposition 4.2, and construct the double curve
sum $S_{z+u}\oplus S_{z-u}$ of $S_{z+u}$ and $S_{z-u}$ which represents $2z$
in $H_{2}(M,\partial M;\mathbb{Z})$. We denote the union of the essential
annuli and the essential disks which come from the double curve sum as $A$. By
Proposition 4.2, $S_{z+u}\oplus S_{z-u}$ is a properly norm-minimizing
surface, and hence we can take a facet surface $F(z)$ of $z$ which contains
$S_{z+u}\oplus S_{z-u}$. By Lemma 5.4, we know that $A$ avoids the guts of
$F(z)$. Because $S_{z+u}$ and $S_{z-u}$ lie in the neighborhood of
$S_{z+u}\oplus S_{z-u}\cup A$, $S_{z+u}$ and $S_{z-u}$ can be isotoped to
avoid the guts of $F(z)$. Thus by lemma 5.1, the image of $z+u$ and $z-u$
under $\varphi$ is zero, which means $\varphi(u)=0$ and $\varphi(z)=0$.

∎

######  Example 5.6.

Denote $\Sigma_{2}$ as a genus 2 closed surface and consider the trivial
circle bundle $\Sigma_{2}\times S^{1}$. Then $H_{2}(\Sigma_{2}\times
S^{1};\mathbb{R})$ is of rank 5 and the Thurston sphere degenerates to 2
parallel hyperplanes. Then for a non-separating closed curve $c_{0}$ in
$\Sigma_{2}$, its fiber bundle $c_{0}\times S^{1}$ is a $T^{2}$ which has 0
Thurston norm. Denote $z$ as the homology class of this $T^{2}$ in
$H_{2}(\Sigma_{2}\times S^{1};\mathbb{R})$. There are 2 non-separating closed
curves $c_{1},c_{2}$ such that $[c_{1}\cup c_{2}]=[c_{0}]$ in
$H_{1}(\Sigma_{2})$ which means $(c_{1}\cup c_{2})\times S^{1}$ is homologous
to $c_{0}\times S^{1}$. If we decompose along $c_{0}\times S^{1}$ and
$(c_{1}\cup c_{2})\times S^{1}$, we have 2 pairs of pants times $S^{1}$. Then
by decomposing along some product annuli, we will have two solid tori with 4
longitudinal sutures where the longitudes are the $S^{1}$ fibers. Hence the
rank of $H_{1}(\Gamma(z))\to H_{1}(\Sigma_{2}\times S^{1})$ is 1.

######  Corollary 5.7 (Cf. [Ago08, Lemma 4.1]).

Let $M$ be a compact irreducible orientable 3-manifold with toral boundary and
non-degenerate Thurston norm. Let

$z\in H^{1}(M)\cong H_{2}(M,\partial M)$

be in a top dimensional open Thurston cone and there is a neighborhood of $z$
which does not contain two elements whose restrictions on a boundary component
are in opposite orientations.

Then we have

$H^{1}(M;\mathbb{Z})\to H^{1}(\Gamma(z);\mathbb{Z})$

is 0, i.e.

$H_{1}(\Gamma(z);\mathbb{Z})\to H_{1}(M;\mathbb{Z})/\textnormal{torsion}$

is 0.

##  6\. Guts and Sutured Decomposition

######  Theorem 1.4.

Let $M$ be an irreducible connected 3-manifold with toral boundary and non-
degenerate Thurston norm. Let $z$ be an element on a $k$-codimensional $(k>0)$
open Thurston face of the Thurston norm. For any choice of $u$ in any
$k^{\prime}$-codimensional $(k^{\prime}<k)$ Thurston face whose closure
contains $z$, there exists a $w$ on the open segment $(z,u)$ such that
$\Gamma(z)$ can be nontrivially decomposed along a properly norm-minimizing
surface representing $w$.

Before proving this theorem, we state a slight generalization of [FK14,
Theorem 4.1].

######  Definition 6.1.

Given two non-zero cohomology classes $\phi,\psi\in H_{2}(M,\partial
M;\mathbb{R})$, we say $\phi$ is _subordinate_ to $\psi$ if $\phi\in C$ where
$C$ is the smallest closed Thurston cone which contains $\psi$.

######  Theorem 6.2.

Let $M$ be an irreducible connected 3-manifold with toral boundary and let $R$
be a properly norm-minimizing surface. Then for any choice of product
decomposition surface (maximal collection of product annuli and product disks)
for $M\backslash R\times(-4,4)$ and any choice of $\psi\in H_{2}(M,\partial
M;\mathbb{Z})$ there exists an $m\in\mathcal{N}$ and a surface $F$ with the
following properties:

  1. (1)

$[R]$ is subordinate to $m[R]+\psi$ and $F$ represents $m[R]+\psi$;

  2. (2)

$F\oplus(R\times\\{-3\\}\cup R\times\\{3\\})$ is a properly norm-minimizing
surface;

  3. (3)

the intersections $F\cap R\times[-4,-2]$ and $F\cap R\times[2,4]$ are product
surfaces;

  4. (4)

if $X$ is a gut or a window of $M\backslash R\times(-4,4)$ then $F\cap X$ is a
decomposition surface.

######  Proof of Theorem 1.4.

By theorem 1.3, $u$ is not in the kernel of the map $\varphi:H^{1}(M)\to
H^{1}(\Gamma(z))$. Let $F(z)$ be a facet surface for $z$ and fix a product
decomposition surface for $M\backslash F(z)\times(-4,4)$.

From Theorem 6.2, we construct a surface $\Sigma$ which represents $mz+u$ such
that $\Sigma$ satisfies the 4 properties. So if we decompose
$M\backslash\backslash(F(z)\times\\{-3\\}\cup F(z)\times\\{3\\})$ along
$\Sigma$, we actually do a sutured decomposition of $M$ along a properly norm-
minimizing surface
$\Omega\stackrel{{\scriptstyle\Delta}}{{=}}\Sigma\oplus(F(z)\times\\{-3\\}\cup
F(z)\times\\{3\\})$ representing some $nz+u$. Since $u$ is not in the kernel
of $\varphi$, we have

$[\Sigma\cap\Gamma(z)]\neq 0\in
H_{2}(\Gamma(z),\partial\Gamma(z);\mathbb{Z}).$

By Theorem 2.18, the complexity of $\Gamma(z)\backslash\Sigma$ is smaller than
$\Gamma(z)$.

Then $mz+u$ is the $w$ that we want in the statement. ∎

##  7\. Examples of Guts

In this section, we give some example of guts in some typical manifolds and
define the guts of knots.

We show an example of guts as solid tori with 2 sutures and $T\times I$ with 2
sutures.

######  Example 7.1.

Book of I-bundles. Denote $\Sigma$ as a compact oriented surface and consider
$\Sigma\times S^{1}$. Let $x_{1},\cdots,x_{k}$ be points in $S^{1}$ and
$\gamma_{i}$ be an essential closed curve in $\Sigma\times\\{x_{i}\\}$ for
$1\leq i\leq k$. Let $\eta(\gamma_{i})$ be the tubular neighborhood of
$\gamma_{i}$, $m_{i}$ be the meridian on $\partial\eta(\gamma_{i})$ which
bounds a disk in $\eta(\gamma_{i})$ and $l_{i}$ be the longitude which is a
component of $(\Sigma\times\\{x_{i}\\})\cap\partial\eta(\gamma_{i})$ such that
$\langle m_{i},l_{i}\rangle=1$. A slope on $\partial\eta(\gamma_{i})$ is of
the form $\pm(q_{i}m_{i}+p_{i}l_{i})$ and is hence determined by the rational
number $\frac{q_{i}}{p_{i}}\in\mathbb{Q}\cup\\{\infty\\}.$

We do a $(q_{i}/p_{i})$-Dehn surgery for each $\gamma_{i}\times\\{x_{i}\\}$
where $|q_{i}|\geq 2$. If we decompose this manifold along a maximal
collection of horizontal surfaces and then decompose along some product annuli
that separate $\gamma_{i}$ from product sutured manifolds, we have $k$ sutured
solid tori as guts components.

Let $m^{\prime}_{i}$ be the meridian of the $i$-th sutured solid torus which
is represented by $q_{i}m_{i}+p_{i}l_{i}$ and
$l^{\prime}_{i}=r_{i}m_{i}+s_{i}l_{i}$ be a longitude satisfying $\langle
m^{\prime}_{i},l^{\prime}_{i}\rangle=1$ which is equivalent to $qs-pr=1$.
Since $\langle m^{\prime}_{i},l_{i}\rangle=q_{i}$ and $\langle
l^{\prime}_{i},l_{i}\rangle=r_{i}$, we have
$l_{i}=-r_{i}m^{\prime}_{i}+q_{i}l^{\prime}_{i}$. Hence the $i$-th solid torus
has two sutures with slope of $-r_{i}/q_{i}$.

If we do drillings along $\gamma_{i}$ instead of Dehn surgery, we will have
$k$ guts which is a $T\times I$ with 2 sutures on a boundary component and the
other component as a sutured torus.

We also show some examples of guts in links complements.

A

B

Figure 10. The Whitehead links.

######  Example 7.2.

Whitehead links. We consider the complement $N$ of the Whitehead links. In
Figure 10, there are 2 Seifert surfaces which are properly norm-minimizing
surfaces representing the homology class related to one link component. The
blue and red Seifert surfaces are parallel and the blue surface is a facet
surface of this link component. Then decomposing along the annuli in orange in
the left of Figure 11, we know that the guts of this facet surface is a
$T\times I$ with 2 sutures on a boundary component and the other component as
a sutured torus.

A

B

Figure 11. A guts component in the complement of the Whitehead links as well
as of the Borromean Rings.

A

B

Figure 12. The Borromean rings.

######  Example 7.3.

Borromean Rings. We consider the complement $N$ of the Borromean Rings. By
[Thu86, Example 2], we know that the Thurston ball is an octahedron and its
vertices are homology classes that are non-trivial on exactly one link
component. In Figure 12, there are 4 Seifert surfaces which are properly norm-
minimizing surfaces representing the homology class $z$ related to one link
component. The blue and green Seifert surfaces are parallel as well as the
yellow and red ones. The blue and red surfaces are a facet surface of this
link component. Then the guts of this facet surface is two pieces of $T\times
I$ with 2 sutures on a boundary component and the other component as a sutured
torus as in Figure 11.

A homology class which is non-trivial on exactly two link components is on an
edge of the Thurston ball. We can construct a norm-minimizing surface $T$
representing such homology class $w$ by constructing Seifert surface for two
link components and cut-and-paste techniques. As stated in Theorem 1.4, we can
conduct a non-trivial decomposition of $\Gamma(z)$ along $T$ so that the guts
of the resulting sutured manifold is a $T\times I$ with 2 sutures on a
boundary component and the other component as a sutured torus.

In the remaining of this section, we analyze the guts of 2-bridge knot
complements.

######  Definition 7.4.

Let $K$ be a knot in $S^{3}$. The _guts_ of $K$ (or the complement of $K$) is
the guts of $z\in H_{2}(S^{3}\backslash K,\partial(S^{3}\backslash K))$ where
$z$ is the homology class of a Seifert surface.

######  Definition 7.5.

A _2-bridge knot_ is a knot which can be regular isotoped so that the natural
height function given by the z-coordinate has only two maxima and two minima
as critical points. For a 2-bridge knot, we can assign a rational number
$p/q$, with $q$ odd such that it is isotoped to a square ”pillowcase” with
slope $\pm p/q$. We denote this 2-bridge knot as $K_{p/q}$. See Figure 13.

Figure 13. Cf. [HT85, Fig. 1]. 2-Bridge Knot $K_{3/5}$.

######  Theorem 7.6 ([HT85, Page 225]).

$K_{p/q}$ is isotopic to $K_{p^{\prime}/q^{\prime}}$ if and only if
$q^{\prime}=q$ and $p^{\prime}\equiv p^{\pm 1}(\textnormal{mod}q)$.

######  Definition 7.7.

We define a _continued fraction_ as

$r+\cfrac{1}{b_{1}-\cfrac{1}{b_{2}-\cfrac{1}{\cdots-\cfrac{1}{b_{k}}}}}$

where $r,b_{i}\in\mathbb{Z}$, and denote it as $r+[b_{1},\ldots,b_{k}]$. We
call $r+[b_{1},\ldots,b_{k}]$ is a _continued fraction expansion_ of $p/q$ if
$p/q=r+[b_{1},\ldots,b_{k}]$.

######  Example 7.8.

$3/5=1+[-2,2]$

Hatcher and Thurston [HT85] classified all incompressible surfaces, and in
particular, oriented Seifert surfaces in 2-bridge knot complements.

######  Theorem 7.9 (Cf. [Con70, Section 1], [HT85, Page 226]).

If $p/q=r+[b_{1},\cdots,b_{k}]$, then $K_{p/q}$ is the boundary of the surface
obtained by plumbing together $k$ bands in a row, the $i$-th band having
$b_{i}$ half-twists (right-handed if $b_{i}>0$ and left-handed if $b_{i}<0$).

There are two essentially different ways of performing each plumbing of two
adjacent bands. One way of describing this choice is to say that instead of
using one of the horizontal plumbing squares, we could use the complement of
this square in the horizontal plane containing it, compactified by a point at
$\infty$. (Thus we are now regarding $S^{3}$ as the 2-point compactification
of $S^{2}\times\mathbb{R}$, with the spheres $S^{2}\times\\{*\\}$ being
horizontal.) See Figure 14.

Figure 14. [AD15, Figure 6.5] Plumbing.

If we include for each plumbing both of these complementary horizontal
plumbing squares, we obtain a certain branched surface
$\Sigma[b_{1},\ldots,b_{k}]$. $\Sigma[b_{1},\ldots,b_{k}]$ carries a large
number of (not necessarily connected) surfaces, labelled
$S_{n}(n_{1},\ldots,n_{k-1})$, where $n\geq 1$ and $0\leq n_{i}\leq n$. By
definition, $S_{n}(n_{1},\ldots,n_{k-1})$ consists of $n$ parallel sheets
running close to the vertical portions of each band of
$\Sigma[b_{1},\ldots,b_{k}]$, which bifurcate into $n_{i}$ parallel copies of
the $i$-th inner plumbing square and $n-n_{i}$ parallel copies of the $i$-th
outer plumbing square. For example, when $n=1$, the surfaces
$S_{1}(n_{1},\ldots,n_{k-1})$ ($n_{i}=0$ or $1$) are just the $2^{k-1}$
plumbings of the original $k$ bands.

######  Proposition 7.10 ([HT85, Proposition 1]).

Orientable incompressible Seifert surfaces for $K_{p/q}$ can be represented by
single-sheeted surfaces $S_{1}(n_{1},\cdots,n_{k-1})$ carried by
$\Sigma[b_{1},\cdots,b_{k}]$ with each $b_{i}$ even.

######  Proposition 7.11.

Let $p/q$ be a rational number with $q$ odd. Then there is only one continued
fraction expansions $p/q=r+[b_{1},\cdots,b_{k}]$ for $p/q$ with each $b_{i}$
even and $|b_{i}|\geq 2$.

###### Proof.

Choose an $r$ such that $p/q-r$ has an even numerator. Then we will always
have different parities on two sides. For the uniqueness, note that
$[b_{1},\cdots,b_{k}]$ has different parities on the numerator and the
denominator, and its absolute value is smaller than 1. ∎

######  Lemma 7.12.

Let $p/q=r+[b_{1},\cdots,b_{k}]$ be the unique continued fraction expansions
with each $b_{i}$ even and $|b_{i}|\geq 2$. Then the guts of $S^{3}-K_{p/q}$
consist of the guts of the complements of $b_{i}$-half-twisted bands in
$S^{3}$.

###### Proof.

We prove this lemma by induction on $k$. When $k=1$, the statement holds
naturally.

Suppose the lemma is true for $k-1$. We take a maximal collection of Seifert
surfaces for $S^{3}-K_{p/q}$

$S_{1}(0,0,\cdots,0),S_{1}(1,0,\cdots,0),S_{1}(1,1,\cdots,0),\ldots,S_{1}(1,1,\cdots,1).$

At the $({k-1})$-th level,
$S_{1}(0,0,\cdots,0,0),S_{1}(1,0,\cdots,0,0),\ldots,S_{1}(1,1,\cdots,1,0)$
have the $i$-th inner plumbing square while $S_{1}(1,1,\cdots,1,1)$ has the
$i$-th outer plumbing square. So locally, it will look like $(k-1)$ pieces of
blue surfaces and 1 red surface in Figure 15. Between each two surfaces, we
can decompose along 4 product disks coming from the plumbing. So altogether we
decompose along the $4(k-1)$ product disks coming from the plumbing. Below the
$(k-1)$-th level, there are $(k-1)$ product sutured manifolds and a complement
of a $b_{k-1}$-half-twisted band in $S^{3}$. Above the $(k-1)$-th level, we
have a product sutured manifolds coming from $S_{1}(1,1,\cdots,1,0)$ and
$S_{1}(1,1,\cdots,1,1)$. If we drop this product sutured manifold, the
remaining sutured manifolds above the $(k-1)$-th level are equivalent to
decomposing $S^{3}-K_{p^{\prime}/q^{\prime}}$ along
$S_{1}(0,0,\cdots,0),S_{1}(1,0,\cdots,0),\ldots,S_{1}(1,1,\cdots,1)$ (only
$(k-2)$ coordinates) where $p^{\prime}/q^{\prime}=r+[b_{1},\cdots,b_{k-1}]$
and then a product disk between each two surfaces on $(k-1)$-th level. Hence
by induction, we know that the lemma holds for any $k$.

Figure 15. [AD15, Figure 6.7] Surfaces bifurcating into inner and outer
plumbing squares.

∎

######  Lemma 7.13.

The complements of a 2-half-twisted band is a product sutured manifold.

###### Proof.

The boundary of a 2-half-twisted band is the Hopf link. The Hopf fibration is
a presentation of $S^{3}$ as an (oriented) circle bundle over $S^{2}$ where
the fibers of the north and south pole in $S^{2}$ is the Hopf link. When one
deletes the Hopf link, one has a circle bundle over $\mathbb{R}\times S^{1}$.
Projecting once more to the circle factor, one has an open annulus bundle over
$S^{1}$ and a fiber is the interior of a 2-half-twisted band. ∎

######  Lemma 7.14.

The complements of a $2k$-half-twisted band is equivalent to a solid torus
with two sutures of slope $1/k$.

###### Proof.

This is because the boundary of a $2k$-half-twisted band is a torus link on a
torus $T^{2}$. We can decompose $S^{3}$ along this torus into two solid torus.
And for the inside solid torus, it is a product sutured manifold. ∎

######  Theorem 7.15.

Let $p/q=r+[b_{1},\cdots,b_{k}]$ be the unique continued fraction expansions
with each $b_{i}$ even and $|b_{i}|\geq 2$. Then the guts of $S^{3}-K_{p/q}$
consist of sutured solid tori with two sutures of slope $2/b_{i}$ for each
$|b_{i}|>2$.

We want to show that the guts contain more information than the Knot Floer
Homology, i.e. the Alexander polynomial and the signature of two bridge knot.

A knot $K$ is called alternating if it admits a planar diagram in which the
over- and under-passes alternate, as we follow the knot. In the case of
alternating knots, knot Floer homology is determined by two classical
invariants, the Alexander polynomial $\Delta_{K}$ and the knot signature
$\sigma(K)$.

######  Theorem 7.16 (Ozsváth-Szabó [OS03]).

Let $K\subset S^{3}$ be an alternating knot with Alexander polynomial
$\Delta_{K}(t)=\sum_{s\in\mathbb{Z}}a_{s}t^{s}$ and signature
$\sigma=\sigma(K)$. Then:

$\widehat{\mathit{HFK}}_{i}(K,s)=\begin{cases}\mathbb{Z}^{|a_{s}|}&\text{if}\
i=s+\tfrac{\sigma}{2},\\\ 0&\text{otherwise.}\end{cases}$

######  Theorem 7.17 ([Min82], [Mur07, Theorem 9.3.6]).

Suppose $K_{p/q}$ is a 2-bridge knot or link with $0<p<q,p\text{ odd, and}$
$\gcd(p,q)=1$. Then

(1)
$\Delta_{K_{p/q}}(t)\doteq\sum_{k=0}^{q-1}(-1)^{k}t^{\sum_{i=0}^{k}\epsilon_{i}}$

where $\epsilon_{i}=(-1)^{\lfloor ip/q\rfloor}$ and $\lfloor x\rfloor$ is the
largest integer less than or equal to $x$.

And the signature, $\sigma(K)$, of K is equal to the number of positive
entries minus the number of negative entries in the sequence of
$\\{\epsilon_{i}\\}$, which is $\sum_{i=0}^{k}\epsilon_{i}$.

We find out a pair of two knots with same Alexander polynomial, signature and
knot Floer homology but having different guts.

######  Example 7.18.

The Alexander polynomial for $K_{11/15}$ is $\Delta_{K_{11/15}}(t)\doteq
4-7t+4t^{2}$ which is the same to the one for $K_{7/15}$. Furthermore, they
have the same signature $3$. However, $11/15=1+[-4,-4]$ and hence by Theorem
7.15, the guts for the $K_{11/15}$ complement are two pieces of sutured solid
tori with two sutures of slope $-1/2$. While since $7/15=1+[-2,-8]$, the guts
for the $K_{7/15}$ complement are a sutured solid torus with two sutures of
slope $-1/4$.

This example shows that the guts contain some new information of knots.

##  8\. Guts and Kakimizu Complex

In this section, we apply the properties of guts to the Kakimizu Complex of a
homology class in a 3-manifold.

We use the same notation as [PS12]. For more detailed description of a
Kakimizu complex of a 3-manifold, one may refer to [PS12].

Let $E$ be a compact connected orientable, irreducible and
$\partial$-irreducible 3-manifold. Let $\gamma$ be a union of oriented
disjoint simple closed curves on $\partial E$, which does not separate any
component of $\partial E$. We fix a class $\alpha$ in the homology group
$H_{2}(E,\partial E,\mathbb{Z})$ satisfying $\partial\alpha=[\gamma]$. A
_spanning surface_ is an oriented surface properly embedded in $E$ in the
homology class $\alpha$ whose boundary is homotopic with $\gamma$.

We now define the simplicial complex $MS(E,\gamma,\alpha)$. The vertex set of
$MS(E,\gamma,\alpha)$ is defined to be $\mathcal{MS}(E,\gamma,\alpha)$, the
set of isotopy classes of spanning surfaces which have minimal genus, i.e.
minimize the Thurston norm. We span an edge on
$\sigma,\sigma^{\prime}\in\mathcal{MS}(E,\gamma,\alpha)$ only if they have
representatives $S\in\sigma,S^{\prime}\in\sigma^{\prime}$ such that the
(connected) lift of $E\setminus S^{\prime}$ to the infinite cyclic cover
associated with $\alpha$ intersects exactly two lifts of $E\setminus S$. In
other words, this means that the _Kakimizu distance_ between $\sigma$ and
$\sigma^{\prime}$ equals one. Simplices are spanned on all complete subgraphs
of the 1–skeleton, i.e. ${MS}(E,\gamma,\alpha)$ is the flag complex spanned on
its 1–skeleton.

######  Definition 8.1.

We call a simplex in the Kakimizu Complex _maximal_ if it can not be properly
contained in another simplex.

The goal of this section is to prove the following theorem.

######  Theorem 8.2.

Let $M$ be an irreducible, orientable $3$-manifold with boundary a disjoint
union of tori $\sqcup_{i=1}^{n}P_{i}$ and non-degenerate Thurston norm. The
dimension of a maximal simplex in a Kakimizu Complex ${MS}(E,\gamma,\alpha)$
is an invariant of the Kakimizu Complex.

Let $\Delta$ be a maximal simplex of the Kakimizu Complex
${MS}(E,\gamma,\alpha)$ and denote its vertices as
$\sigma_{0},\ldots,\sigma_{n}$. As in [PS12, Section 3], we can choose
representatives $S_{0},\ldots,S_{n}$ of the vertices such that each two
surfaces are almost transverse and have simplified intersection. Therefore the
distance between each two surfaces is 1 and $S_{0},\ldots,S_{n}$ are pairwise
disjoint. Let $S$ be the union of $S_{0},\ldots,S_{n}$.

Denote $\widetilde{M}$ as the infinite cyclic cover of $M$ associated with the
(kernel of the) element of $H^{1}(M,\mathbb{Z})$ dual to $\alpha$. Let
$\pi\colon\widetilde{M}\rightarrow M$ be the covering map. Let $\tau$ be the
generator of the group of covering transformations of $\widetilde{M}$. Pick a
base point $x_{0}$ disjoint with $S$ and let $\tilde{x}_{0}$ be a lift of
$x_{0}$. The hypothesis that $\gamma$ does not separate the components of
$\partial M$ guarantees that
$N_{i}\stackrel{{\scriptstyle\Delta}}{{=}}M\setminus S_{i}$ is connected. Let
$\tilde{N}^{0}_{i}$ denote the lift of $N_{i}$ to $\widetilde{M}$ that
contains $\tilde{x}_{0}$ and denote
$\tilde{N}^{j}_{i}=\tau^{j}(\tilde{N}^{0}_{i})$ for $j\in\mathbb{Z}$. Denote
also
$\tilde{S}^{j}_{i}=\overline{\tilde{N}^{j-1}_{i}}\cap\overline{\tilde{N}^{j}_{i}}$
for $j\in\mathbb{Z}$.

For a point $q$ in $\tilde{M}$ that is disjoint with the lifts of $S_{i}$, we
define the _potential_ function $\phi_{i}(q)$ to be the index $j$ of
$\tilde{N}^{j}_{i}$ which contains $q$. We denote $\tilde{S}$ as the union of
lifts of $S$ and let $\Phi(q)=\sum_{i=0}^{n}\phi_{i}(q)$ for $q$ in the
complements of $\tilde{S}$.

######  Lemma 8.3.

If $\Phi(q)=\Phi(q^{\prime})$, then $\phi_{i}(q)=\phi_{i}(q^{\prime})$ for
$0\leq i\leq n$.

###### Proof.

Suppose $\Phi(q)=\Phi(q^{\prime})$ and the equalities
$\phi_{i}(q)=\phi_{i}(q^{\prime})$ do not hold for all $i$. Then there exist
$i,i^{\prime}$ such that

$\phi_{i}(q)<\phi_{i}(q^{\prime})$

and

$\phi_{i^{\prime}}(q)>\phi_{i^{\prime}}(q^{\prime}).$

So $\phi_{i}(q^{\prime})=\phi_{i}(q)+k$ and
$\phi_{i^{\prime}}(q^{\prime})=\phi_{i^{\prime}}(q)-k^{\prime}$ where
$k,k^{\prime}\geq 1$. Denote $j$ and $j^{\prime}$ as $\phi_{i}(q)$ and
$\phi_{i^{\prime}}(q)$. Then $\tilde{N}^{j}_{i}$ intersects
$\tilde{N}^{j^{\prime}}_{i^{\prime}}$ and $\tilde{N}^{j+k}_{i}$ intersects
$\tilde{N}^{j^{\prime}-k^{\prime}}_{i^{\prime}}$. By applying the covering
transformation $\tau$ on $\tilde{N}^{j+k}_{i}$ and
$\tilde{N}^{j^{\prime}-k^{\prime}}_{i^{\prime}}$, we know that
$\tilde{N}^{j}_{i}$ intersects
$\tilde{N}^{j^{\prime}-k^{\prime}-k}_{i^{\prime}}$. Therefore,
$\tilde{N}^{j}_{i}$ intersects both $\tilde{N}^{j^{\prime}}_{i^{\prime}}$ and
$\tilde{N}^{j^{\prime}-k^{\prime}-k}_{i^{\prime}}$. This means the distance
between $S_{i}$ and $S_{i}^{\prime}$ is at least $k+k^{\prime}\geq 2$ (see
[PS12, Definition 2.1]), which is a contradiction.

∎

######  Lemma 8.4.

If $\Phi(q^{\prime})=\Phi(q)+1$, then there exists an $i_{0}$ such that
$\phi_{i_{0}}(q^{\prime})=\phi_{i_{0}}(q)+1$ and
$\phi_{i}(q^{\prime})=\phi_{i}(q)$ for $i\neq i_{0}$.

###### Proof.

Since $\tilde{M}$ is connected, there is a path $\eta$ connecting $q$ and
$q^{\prime}$, and furthermore $\eta$ is transverse to the lifts of $S$. We
know that when $\eta$ passes a component of lifts of $S$, the value of $\Phi$
is increased or decreased by 1. So there exist a pair of points $p$ and
$p^{\prime}$ on $\eta$ such that $\Phi(p)=\Phi(q)$,
$\Phi(p^{\prime})=\Phi(q^{\prime})$. Furthermore, $p$ and $p^{\prime}$ is
separated by a $S_{i_{0}}^{j+0}$. Then
$\phi_{i_{0}}(p^{\prime})=\phi_{i_{0}}(p)+1$ and
$\phi_{i}(p^{\prime})=\phi_{i}(p)$ for $i\neq i_{0}$.

By Lemma 8.3, we know that $\phi_{i}(q)=\phi_{i}(p)$ and
$\phi_{i}(q^{\prime})=\phi_{i}(p^{\prime})$ for all $0\leq i\leq n$. Hence the
equalities in the lemma hold. ∎

Now we start the proof of Theorem 8.2.

###### Proof of Theorem 8.2.

By the preceding lemmas, for each $k\in\mathbb{Z}$, $\Phi(q)=k$ determines
unique solutions of $\\{\phi_{i}(q)\\}$. Furthermore, increasing $\Phi(q)$ by
1 will result in increasing a $\phi_{i_{0}}(q)$ by 1. We call this property
the _monotonicity_ of $\Phi$.

Denote $U(k)$ as $\\{q|\Phi(q)=k\\}$. Then $\cup_{k\in
Z}U(k)=\tilde{M}\setminus\tilde{S}$ and
$\overline{U(k)}\cap\overline{U(k^{\prime})}\neq\emptyset$ if and only if
$|k-k^{\prime}|=1$. Furthermore for an integer $k_{0}$,
$\overline{U(k_{0})}\cap\overline{U(k_{0}+1)}$ is a subset of
$\tilde{S}_{i_{0}}^{j_{0}}$ by Lemma 8.4. Note that
$\tilde{S}_{i_{0}}^{j_{0}}$ is
$\overline{\tilde{N}^{j_{0}-1}_{i_{0}}}\cap\overline{\tilde{N}^{j_{0}}_{i_{0}}}$.
From the monotonicity of $\Phi$, we know that only one $k$ satisfies that
$\overline{U(k)}\cap\overline{U(k+1)}$ is a subset of
$\tilde{S}_{i_{0}}^{j_{0}}$. Hence
$\overline{U(k_{0})}\cap\overline{U(k_{0}+1)}$ is $\tilde{S}_{i_{0}}^{j_{0}}$.

By arranging the order of $S_{i}$, we can let
$\overline{U(0)}\cap\overline{U(1)}$ be $\tilde{S}_{0}^{1}$,
$\overline{U(1)}\cap\overline{U(2)}$ be $\tilde{S}_{1}^{1}$, …,
$\overline{U(n)}\cap\overline{U(n+1)}$ be $\tilde{S}_{n}^{1}$. By applying the
projection map $\pi$ on $\overline{U(i)}$ for $0\leq i\leq n$, we denote
${\mathcal{L}}_{i}=\pi(\overline{U(i)})$. Then
${\mathcal{L}}_{i}\cap{\mathcal{L}}_{i+1}=S_{i}$ (mod $(n+1)$) for $0\leq
i\leq n$.

We say ${\mathcal{L}}_{i}$ is the $i$-th _layer_ and think of it as a sutured
manifold with $R_{-}=S_{i}$ and $R_{+}=S_{i+1}$. We call the combination of
layers a _sutured bundle_ structure for $M$ with respect to $S$.

Since $S_{i}$ is not isotopic to $S_{i^{\prime}}$ for $i\neq i^{\prime}$, each
${\mathcal{L}}_{i}$ is not a product sutured manifold.

Suppose there is a properly norm-minimizing surface $T$ such that
$[T]=\alpha$, $T$ is disjoint with $S$ and a component of $T$ is not parallel
to $S$. Without loss of generality, we assume that component is in
${\mathcal{L}}_{0}$. Let $T^{\prime}$ denote $T\cap{\mathcal{L}}_{0}$. Since
$T$ is homologous to $S_{0}$, there is a region $N$ of $M$ such $\partial N$
is $T^{-}\cup S_{0}$ where $T^{-}$ comes from reversing the orientation of
$T$. Then the boundary of $N\cap{\mathcal{L}}_{0}$ is $(T^{\prime})^{-}\cup
S^{\prime}_{0}$ where $S^{\prime}_{0}$ is the union of some components of
$S_{0}$. Hence $T^{\prime}$ is homologous to $S^{\prime}_{0}$. Denote $T_{0}$
as $(S_{0}-(S^{\prime}_{0}))\cup T^{\prime}$ and perturb $T_{0}$ in the
neighborhood of $S_{0}$ so that $T_{0}$ is in the interior of $N$. Since
$T^{\prime}$ is not parallel to $S$, we know that $T_{0}$ represents a vertex
of ${MS}(E,\gamma,\alpha)$ different from $S_{0},\ldots,S_{n}$. By adding
$T_{0}$, we have a sutured bundle with $(n+2)$ layers and hence the distance
between $T_{0}$ and $S_{i}$ is 1 for $0\leq i\leq n$. Therefore
$T_{0},S_{0},\ldots,S_{n}$ forms a simplex contains the maximal simplex
$\Delta$ which is a contradiction.

Let $F$ be a facet surface contains $S$. From the above, we know that
$M\backslash\backslash F$ is the union of $M\backslash\backslash S$ and some
product sutured manifolds. Therefore the guts $\Gamma(M,F)$ of
$M\backslash\backslash F$ is the guts of $M\backslash\backslash S$. By Lemma
3.7, the guts of ${\mathcal{L}}_{i}$ is connected, i.e. a component of the
guts of $\Gamma(M,F)$. Hence the number of components of $\Gamma(M,F)$ is
$n+1$. In other words, the dimension $n$ of the maximal simplex $\Delta$ is
$|\Gamma(M,F)|-1$. By Theorem 1.1 and Definition 3.18, $n$ is equal to
$|\Gamma(\alpha)|-1$, which does not depend on the selection of the maximal
simplex $\Delta$.

∎

##  9\. Conclusion

The main result in this paper was motivated by understanding the relation
between the Thurston norm and sutured manifolds. In a sequel, this will be
applied to understand the minimal volume 3-cusped hyperbolic manifolds. As
described before, it will be interesting to see the ramifications for the
structure of the Kakimizu complex.

The main theorem in this paper should generalize to give guts for homology
classes of sutured manifolds. There is a polytope associated to the homology
of a sutured manifold, and we expect that these guts will have similar
relations with this polytope.

Another possible direction that the main results of this paper may be useful
for is to understand the structure of the Thurston norm polytope. In
particular, it is unknown for betti number $>2$ which polytopes can be the
unit ball of the Thurston norm of a 3-manifold? The local structure of the
polytope near a vertex should be governed by the polytope of the sutured
manifold of the face associated to that point. But it is harder to imagine how
this local structure could be integrated to get a global picture of the
polytope.

The Example 7.18 indicates that the guts contains more information than
sutured Floer homology. One may ask whether one can incorporate more
information to refine sutured Floer homology? In particular, are there
homology operations on Floer homology that might distinguish sutured solid
tori with one or more components?

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