# Morse-Novikov numbers, tunnel numbers, and handle numbers of sutured
manifolds

Kenneth L. Baker  Department of Mathematics, University of Miami, Coral
Gables, FL 33146, USA<EMAIL_ADDRESS>and  Fabiola Manjarrez-Gutiérrez
Instituto de Matemáticas, Universidad Nacional Autónoma de Mexico, Cuernavaca,
Mor., MEXICO<EMAIL_ADDRESS>
###### Abstract.

Developed from geometric arguments for bounding the Morse-Novikov number of a
link in terms of its tunnel number, we obtain upper and lower bounds on the
handle number of a Heegaard splitting of a sutured manifold $(M,\gamma)$ in
terms of the handle number of its decompositions along a surface representing
a given 2nd homology class. Fixing the sutured structure $(M,\gamma)$, this
leads us to develop the handle number function $h\colon H_{2}(M,\partial
M;\mathbb{R})\to\mathbb{N}$ which is bounded, constant on rays from the
origin, and locally maximal. Furthermore, for an integral class $\xi$,
$h(\xi)=0$ if and only if the decomposition of $(M,\gamma)$ along some surface
representing $\xi$ is a product manifold.

###### Key words and phrases:

sutured manifolds, handle number, Heegaard splittings, Morse-Novikov number,
tunnel number

###### 2020 Mathematics Subject Classification:

57K10, 57K35, 57K99

###### Contents

  1. 1 Introduction
    1. 1.1 Bounds between linear and circular Heegaard splittings of sutured manifolds
    2. 1.2 Handle indices
    3. 1.3 A function on homology
    4. 1.4 Acknowledgements
  2. 2 Terminology
    1. 2.1 Notations and conventions
    2. 2.2 Sutured manifolds and compression bodies
    3. 2.3 Handle numbers, Heegaard splittings
    4. 2.4 Circular Heegaard splittings
    5. 2.5 Product decompositions and handle numbers
    6. 2.6 Handle structures, handle counts, and handle indices
  3. 3 Some operations on Heegaard splittings
    1. 3.1 Amalgamations
    2. 3.2 Realization by homologically essential surfaces
    3. 3.3 Conditioned splittings
    4. 3.4 Tubings of splittings
  4. 4 Restructuring sutures
  5. 5 An upper bound for sutured manifolds without toroidal sutures
    1. 5.1 Decompositions of compression bodies
    2. 5.2 An Upper Bound
  6. 6 A lower bound for sutured manifolds without toroidal sutures
    1. 6.1 Sutured Characteristic
    2. 6.2 A Lower Bound
  7. 7 Bounds for knots and links
    1. 7.1 Tunnel numbers and Morse Novikov numbers of links
    2. 7.2 Bounds for knots and links in $S^{3}$.
  8. 8 The circular handle number function on homology
    1. 8.1 Regular classes
    2. 8.2 A function on homology
  9. 9 Further remarks and questions.
    1. 9.1 Regarding handle numbers and faces of the Thurston norm
    2. 9.2 Handle complexities of 2nd homology classes

##  1\. Introduction

Let $L$ be an oriented link in $S^{3}$ with Morse-Novikov number
${\rm{MN}}(L)$ and tunnel number ${\rm{TN}}(L)$. Letting
$M=S^{3}\backslash\mathcal{N}(L)$ be the link exterior, $2{\rm{TN}}(L)+2$
counts the minimum number of critical points of a Morse function on $M$ for
which $\partial M$ is the maximal level while ${\rm{MN}}(L)$ counts the
minimum number of critical points among regular circle valued Morse functions
on $M$ dual to the meridional class. In [Paj10] A. Pajitnov proved that
${\rm{MN}}(L)\leq 2{\rm{TN}}(L)$ using a perturbation argument to construct a
circle-valued Morse map from a given real valued Morse function on the link
exterior. Discussions about a constructive geometric argument for this upper
bound on ${\rm{MN}}(L)$ led to developing a lower bound and further
generalizations for sutured manifolds.

###  1.1. Bounds between linear and circular Heegaard splittings of sutured
manifolds

For our constructive geometric arguments, we shift from Morse functions and
circular Morse functions to handle decompositions, Heegaard splittings, and
circular Heegaard splittings where we count handles instead of critical
points.

Given a connected sutured manifold $(M,\gamma)$ without toroidal sutures, a
Heegaard surface is a connected decomposing surface $S$ for which $\partial S$
is the union of the cores of the annular sutures $A(\gamma)$ so that
$(M,\gamma)$ decomposed along $S$ yields a pair of connected compression
bodies for which $S$ is the “thick” surface and any annular suture is a
vertical boundary component. The surface $S$ defines a linear Heegaard
splitting of $(M,\gamma)$.

More generally, for a “thin” decomposing surface $F$ for $(M,\gamma)$ that
meets every toroidal suture, let the “thick” surface $S$ be the union of a
Heegaard surface for each component of the decomposition of $(M,\gamma)$ by
$F$. Then together $F$ and $S$ define a circular Heegaard splitting of
$(M,\gamma)$. Note that a linear Heegaard splitting is simply a circular
Heegaard splitting with no thin surface, $F=\emptyset$.

Dropping $\gamma$ from the notation, the handle number $h(M,F,S)$ of the
splitting of $(M,\gamma)$ with the thin and thick surfaces $F$ and $S$ counts
the minimum number of handles needed to create the compression bodies in the
decomposition of $(M,\gamma)$ first along $F$ and then along $S$. Then we
define the handle number $h(M,\gamma,\xi)$ of a class $\xi\in H_{2}(M,\partial
M)$, to be the minimum of $h(M,F,S)$ among circular Heegaard splittings for
which $[F]=\xi$. Of course this is only defined for classes $\xi$ represented
by decomposing surfaces that meet every toroidal suture, and we say such
classes are regular. This regularity is a kind of “non-zero linking number”
condition.

When $(M,\gamma)$ has no toroidal sutures, every class is regular. In
particular $\xi=0$ is a regular class and the linear handle number,
$h(M,\gamma,0)$, is realized by a linear Heegaard splitting with
$F=\emptyset$.

Moreover, as we require neither the thin surface $F$ nor the result of the
decomposition of $(M,\gamma)$ along $F$ to be connected, our circular Heegaard
splittings $(M,F,S)$ may actually be what are commonly called generalized
Heegaard splittings or generalized circular Heegaard splittings according to
whether or not the homology class of $F$ is trivial. For simplicity, we will
just call them all Heegaard splittings, using the adjectives linear, circular,
and generalized for emphasis where useful. Indeed, note that a Heegaard
splitting $(M,F,S)$ in which $[F]$ is a non-primitive homology class is
necessarily a generalized circular Heegaard splitting.

Finally, we also define a sort of Euler characteristic complexity
$\chi^{h}(M,\gamma,\xi)=\chi^{h}_{(M,\gamma)}(\xi)$ that measures the minimum
“size” of thin surface needed in a Heegaard splitting that realizes the handle
number of the class $\xi$. This complexity arises naturally when we bound the
circular handle number from below by the linear handle number.

######  Theorem 6.6.

Suppose $(M,\gamma)$ is a sutured manifold without toroidal sutures. Then for
any $\xi\in H_{2}(M,\partial M)$ we have

$h(M,\gamma,\xi)\leq h(M,\gamma,0)-2d\leq
h(M,\gamma,\xi)+2\chi^{h}_{(M,\gamma)}(\xi)$

where $d=0,1,2$ depending on whether neither, just one, or both of
$R_{+}(\gamma)$ and $R_{-}(\gamma)$ are empty.

###### Proof.

Theorem 5.4 develops the first inequality while Theorem 6.5 develops the
second. ∎

As an application, let $(M,\gamma_{+})$ be the sutured manifold where
$M=S^{3}\backslash\mathcal{N}(L)$ is the exterior of an oriented link $L$ in
$S^{3}$ with $R_{+}(\gamma_{+})=\partial M$. Let $\xi\in H_{2}(M,\partial M)$
be the class of a Seifert surface for $L$. Then
${\rm{MN}}(L)=h(M,\gamma_{+},\xi)$ while $2{\rm{TN}}(L)+2=h(M,\gamma_{+},0)$.
Since $R_{+}(\gamma_{+})=\partial M\neq\emptyset$ while
$R_{-}(\gamma_{+})=\emptyset$, the bound

$h(M,\gamma,\xi)\leq h(M,\gamma,0)-2d$

of Theorem 6.6 reduces to Pajitnov’s bound ${\rm{MN}}(L)\leq 2{\rm{TN}}(L)$,
[Paj10]. For the lower bound on ${\rm{MN}}(L)$ in this context, let us define
the Morse-Novikov genus $g_{{\rm{MN}}}(L)$ to be the minimum genus among
(possibly disconnected) Seifert surfaces $F$ for $L$ that are the thin surface
of a Heegaard splitting realizing ${\rm{MN}}(L)=h(M,\gamma_{+},\xi)$. It turns
out that in this situation $\chi^{h}(M,\gamma_{+},\xi)$ is just
$2g_{{\rm{MN}}}(L)$. Hence Theorem 6.6 becomes

######  Theorem 7.6.

For an oriented link $L$ in $S^{3}$,

${\rm{MN}}(L)\leq 2{\rm{TN}}(L)\leq{\rm{MN}}(L)+4g_{{\rm{MN}}}(L).$

######  Corollary 7.7.

If $L$ is a fibered link, then ${\rm{TN}}(L)\leq 2g(L)$.

###### Proof.

If $L$ is fibered, then ${\rm{MN}}(L)=0$ and $g_{\rm{MN}}(L)=g(L)$ since there
is a unique incompressible Seifert surface. Then Theorem 7.6 implies the
result. ∎

One may expect that $g_{{\rm{MN}}}(L)=g(L)$, that the Morse-Novikov genus of
an oriented link $L$ is realized by a minimal genus Seifert surface. However
it is not so clear. Nevertheless, by [Bak21], we do know the Morse-Novikov
genus of a knot is realized by an incompressible Seifert surface. That proof
also extends to oriented links allowing for disconnected Seifert surfaces.

######  Conjecture 7.12 (Cf. [MG09, Remark p440]).

For an oriented link $L$ in $S^{3}$, $g_{{\rm{MN}}}(L)=g(L)$.

In Section 9.2, Question 9.6 asks about similar relationships for more general
sutured manifolds and homology classes.

###  1.2. Handle indices

Theorem 6.6 has a somewhat cleaner statement using handle indices rather than
handle numbers. Instead of adding up the handles of all dimensions, the handle
index counts the number of $1$ and $2$ handles minus the number of $0$ and $3$
handles. Effectively, the handle index $j(M,F,S)$ of a splitting is the handle
number $h(M,F,S)$ minus twice the number of handlebodies in the decomposed
manifold. It turns out that at most one $0$–handle and one $3$–handle is used
in splittings realizing $h(M,\gamma,\xi)$, corresponding directly to whether
$R_{-}(\gamma)=\emptyset$ or $R_{+}(\gamma)=\emptyset$; see Lemma 5.2.
Consequently $j(M,\gamma,\xi)=h(M,\gamma,\xi)$ for all $\xi\neq 0$ while
$j(M,\gamma,0)=h(M,\gamma,0)-2d$ for $d$ as in Theorem 6.6. Hence Theorem 6.6
may be rewritten as:

######  Corollary 1.1.

Suppose $(M,\gamma)$ is a sutured manifold without toroidal sutures. Then for
any $\xi\in H_{2}(M,\partial M)$ we have

$j(M,\gamma,\xi)\leq j(M,\gamma,0)\leq
j(M,\gamma,\xi)+2\chi^{h}(M,\gamma,\xi)$

Written as such, this prompts one to examine the relationships among
$h(M,\gamma,\xi)$ over all regular classes $\xi\in H_{2}(M,\partial M)$.

###  1.3. A function on homology

The arguments that prove Theorem 6.6 reveal interesting structure of
$h(M,\gamma,\xi)$ as a function of the regular classes of $H_{2}(M,\partial
M)$. Here we fix the sutured manifold $(M,\gamma)$ and write $h(\xi)$ for
$h(M,\gamma,\xi)$, viewing it as a function on homology.

######  Theorem 8.3.

For a sutured manifold $(M,\gamma)$, the handle number function

$h\colon H_{2}(M,\partial M)\to\mathbb{N}$

has the following properties for any regular classes $\alpha,\beta\in
H_{2}(M,\partial M)$:

  1. (1)

$h(\beta)\leq N$ for some constant $N$ (depending on $(M,\gamma)$),

  2. (2)

$h(k\beta)=h(\beta)$ for any integer $k>0$, and

  3. (3)

$h(\alpha+m\beta)\leq h(\beta)$ for sufficiently large $m$.

As the regular classes form a dense open set of $H_{2}(M,\partial M)$ as shown
in Lemma 8.2, these properties enable a natural extension to a function on
homology with real coefficients.

######  Theorem 8.4.

The handle number extends to a function

$h\colon H_{2}(M,\partial M;\mathbb{R})\to\mathbb{N}$

that is

  1. (1)

bounded by constant,

  2. (2)

constant on rays from the origin, and

  3. (3)

locally maximal.

As discussed in Section 9.1, if $(M,\gamma)$ is a sutured manifold with
$\partial M$ toroidal and no annular sutures, then $h=0$ exactly on fibered
faces of the Thurston norm. More generally, for an integral class $\xi$,
$h(\xi)=0$ if and only if the decomposition of $(M,\gamma)$ along some surface
representing $\xi$ is a product manifold.

###  1.4. Acknowledgements

KLB was partially supported by a grant from the Simons Foundation (grant
#523883 to Kenneth L. Baker).

FM-G was partially supported by Grant UNAM-PAPIIT IN108320.

##  2\. Terminology

Much of this can be found throughout the literature in various forms, though
we take some liberty with resetting terminology so as to not be encumbered
with modifiers. We largely follow and broaden the perspective in [Bak21] based
on work of Goda [God06].

###  2.1. Notations and conventions

All manifolds are assumed to be compact and oriented. All submanifolds are
assumed to be properly embedded. If $Y$ is a $n$–manifold and $S\subset Y$ is
a properly embedded submanifold we denote by $\mathcal{N}(S)$ an open regular
neighborhood around $S$, and by $\mathcal{N}(\partial S)$ the open regular
neighborhood $\mathcal{N}(S)\cap\partial Y$ of $\partial S$ in $\partial Y$.

If the codimension of $S$ in $Y$ is one, then $\mathcal{N}(S)$ is called a
(bi)collar neighborhood of $S$ and is equivalent to
$S\times(-\epsilon,\epsilon)$, for $\epsilon>0$. The product
$S\times[-\epsilon,\epsilon]$, for $\epsilon>0$, is called a closed collar
neighborhood of $S$, or simply a collar of $S$.

We use $Y\backslash S$ to denote $Y-\mathcal{N}(S)$.

###  2.2. Sutured manifolds and compression bodies

Sutured manifolds were introduced by Gabai [Gab83] and provide a convenient
framework for discussing compression bodies and various generalizations of
Heegaard splittings, e.g. [God06, Bak21]

A sutured manifold is a compact oriented $3$–manifold $M$ with a disjoint pair
of subsurfaces $R_{+}$ and $R_{-}$ of $\partial M$ such that

  * •

the orientation of $R_{+}$ is consistent with the boundary orientation of
$\partial M$ while the orientation of $R_{-}$ is reversed,

  * •

$\partial M\backslash(R_{+}\cup R_{-})$ is a collection $\gamma$ of annuli
$A(\gamma)$ and tori $T(\gamma)$, collectively called the sutures, and

  * •

each annular suture joins a component of $\partial R_{+}$ to a component of
$\partial R_{-}$.

Typically, sutured manifolds are written as $(M,\gamma)$ where $\gamma$
represents the sutured structure. With this sutured structure understood, we
also may use $\gamma$ to represent the core curves of $A(\gamma)$, oriented to
be isotopic to $\partial R_{+}$. At times it is convenient to use
$\partial_{+}M$ and $\partial_{-}M$ to refer to the surfaces $R_{+}$ and
$R_{-}$. (One may also care to use $\partial_{v}M$, for vertical boundary), to
refer to the annular sutures $A(\gamma)$.) We may also use variations of
$R_{\pm}(M)$ or $R_{\pm}(\gamma)$ for these surfaces $R_{-}$ and $R_{+}$.

Suppose $(S,\partial S)$ and $(T,\partial T)$ are two embedded oriented
surfaces in general position in $(M,\partial M)$. Define
$T\mathbin{\rotatebox[origin={c}]{-30.0}{$\asymp$}}S$ to be the double curve
sum of $T$ and $S$, i.e., cut and paste along the curves of intersection to
get an embedded oriented surface representing the cycle $T+S$. The surface
$T\mathbin{\rotatebox[origin={c}]{-30.0}{$\asymp$}}S$ is an embedded oriented
surface coinciding with $T\cup S$ outside a regular neighborhood of the
intersections $T\cap S$. See [Sch89, Definition 1.1].

A decomposing surface $(S,\partial S)\subset(M,\partial M)$ is an oriented
properly embedded surface which intersects each toroidal suture in coherently
oriented parallel essential circles and each annular suture either in circles
parallel to the core of the annulus or in essential arcs (not necessarily
oriented coherently). There is a natural sutured manifold structure
$(M^{\prime},\gamma^{\prime})$ on $M^{\prime}=M\backslash S$, where:

  * •

$\gamma^{\prime}=(\gamma\cap M^{\prime})\cup\mathcal{N}(S_{+}\cap
R_{-})\cup\mathcal{N}(S_{-}\cap R_{+})$,

  * •

$R_{+}^{\prime}=((R_{+}\cap M^{\prime})\cup S_{+})-int(A(\gamma^{\prime}))$,
and

  * •

$R_{-}^{\prime}=((R_{-}\cap M^{\prime})\cup S_{-})-int(A(\gamma^{\prime}))$.

Here $\mathcal{N}(S)=S\times(-\epsilon,\epsilon)$ is a bicollar neighborhood
of $S$ where $S=S\times\\{0\\}$ and we set $S_{\pm}=S\times\\{\mp\epsilon\\}$.
With this arrangement, the normal of the oriented surface $S_{+}$ points out
of $M\backslash S$ and of $S_{-}$ points into $M^{\prime}=M\backslash S$. We
say $(M^{\prime},\gamma^{\prime})$ is obtained from a sutured manifold
decomposition of $(M,\gamma)$ along $S$, and we denote this decomposition as
$(M,\gamma)\overset{S}{\leadsto}(M^{\prime},\gamma^{\prime})$. See Figure 2.

Figure 1. The double curve sum
$S\mathbin{\rotatebox[origin={c}]{-30.0}{$\asymp$}}T$ of the surfaces $S$ and
$T$. Figure 2. A sutured manifold decomposition along a surface $S$.

A compression body $C$ is a special kind of sutured manifold that may be
formed from a collar of $R_{-}$ and a collection of $0$–handles by attaching
$1$–handles. The collar of $R_{-}$ is a product $R_{-}\times[-1,1]$ of a non-
empty and possibly disconnected compact oriented surface $R$. Then $1$–handles
are attached to $R_{-}\times\\{1\\}$ and the boundaries of the $0$–handles. As
a sutured manifold, the surface $\partial_{-}C=R_{-}$ is identified with
$R_{-}\times\\{-1\\}$, the annular sutures are $\partial_{v}C=\partial
R_{-}\times[-1,1]$, and $\partial_{+}C=R_{+}$ is what remains of $\partial C$.

A meridional disk of a compression body is a compressing disk for $R_{+}$. In
the above formulation of compression bodies, $R_{-}$ is necessarily
incompressible in the compression body. For such a compression body $C$, we
may say that $\partial_{-}C=R_{-}$ is the thin boundary, $\partial_{+}C=R_{+}$
is the thick boundary, and $\partial_{v}C=A(\gamma)$ is the vertical boundary.

A trivial compression body is a compression body without any handles; hence it
is a product. A product ball is a trivial compression body homeomorphic to a
$3$–ball when the sutured manifold structure is forgotten; it has the form
$D^{2}\times[-1,1]$.

A handlebody is a connected compression body in which $R_{-}=\emptyset$. The
entire boundary is $R_{+}$, a closed oriented surface. Hence, a handlebody may
be viewed as a single $0$–handle with $g$ $1$–handles attached where $g$ is
the genus of $R_{+}$. We also say $g$ is the genus of the handlebody. More
generally one may regard a handlebody as a closed regular neighborhood of a
connected finite graph in which the vertices give rise to the collection of
$0$–handles and the edges determine the $1$–handles.

We may “dualize” a compression body by swapping $R_{+}$ and $R_{-}$. Then a
dualized compression body may be regarded as a sutured manifold formed from a
collar of $R_{-}$ by attaching $2$–handles and filling with $3$–handles a
collection of resulting spherical boundary components that are disjoint from
the vertical boundary. In passing between a compression body and a dualized
compression body, duality exchanges the $1$– and $2$–handles, the $0$– and
$3$–handles, and the roles of the surfaces $R_{+}$ and $R_{-}$. So, dually,
$R_{+}$ is the thin boundary, $R_{-}$ is the thick boundary, and a handlebody
is a connected dualized compression body in which $R_{+}=\emptyset$. See
Figure 3.

Figure 3. A compression body and a dualized compression body, each shown first
with a schematic depiction and then with an example. The examples also show
the cores for a choice of $1$–handles and $2$–handles.

######  Remark 2.1.

Forgetting about the gradient flows implied by the indices of handles and the
(normal) orientations of $R_{+}$ and $R_{-}$, a compression body $C$ can be
viewed as a $3$–manifold obtained by

  * •

(the “Inside-Out” construction) attaching $1$–handles to a collar of the
surface $\partial_{-}C$ and some $0$–handles so that no spherical boundary is
left disjoint from $\partial_{v}C$, or equivalently

  * •

(the “Outside-In” construction) attaching $2$–handles to a collar of the
surface $\partial_{+}C$ and filling with $3$–handles a collection of spherical
boundary components disjoint from $\partial_{v}C$.

###  2.3. Handle numbers, Heegaard splittings

The handle number $h(C)$ of a compression body $C$ is the minimum number of
$0$– and $1$–handles needed to form $C$. The handle number of a handlebody $H$
of genus $g$ is $h(H)=g+1$. In the dualized setting, the handle number is the
minimum number of $2$– and $3$–handles needed to form the compression body.

Let $(M,\gamma)$ be a sutured manifold without toroidal sutures. A linear
Heegaard splitting of $M$, denoted $(M,S)$ or $(A,B;S)$, is a decomposition of
$M$ into a compression body $A$ and dual compression body $B$ along a surface
$S=R_{+}(A)=R_{-}(B)$.

Observe that $S$ is then a decomposing surface with $\partial S$ contained in
the annular sutures of $M$. Then the sutured manifold decomposition
$M^{\prime}=M\backslash S$ yields a disjoint union of a compression body
$A^{\prime}$ and a dualized compression body $B^{\prime}$ where
$S_{+}=R_{+}(A^{\prime})$, $S_{-}=R_{-}(B^{\prime})$, and
$M=A^{\prime}\cup\mathcal{N}(S)\cup B^{\prime}$. In particular,
$A=A^{\prime}\cup S\times[-\epsilon,0]$ and
$B=B^{\prime}\cup\times[0,\epsilon]$ so that $A\cong A^{\prime}$ and $B\cong
B^{\prime}$ as sutured manifolds. Note that $R_{-}(M)=R_{-}(A)$ and
$R_{+}(M)=R_{+}(B)$. See Figure 4.

The surface $S$ is called a Heegaard surface for $(M,\gamma)$. As $S$ is the
thick boundary of both $A$ and $B$, we also refer to $S$ as a thick surface.
The Heegaard genus $g(M,\gamma)$ of the sutured manifold $(M,\gamma)$ is the
minimum genus of its Heegaard surfaces.

######  Remark 2.2.

Traditionally, the term Heegaard splitting is used for a decomposition of a
manifold $M$ along a connected surface $S$ into either two handlebodies when
$\partial M=\emptyset$ or two connected compression bodies without vertical
boundary when $\partial M\neq\emptyset$. We include such splittings among
linear Heegaard splittings. Ultimately, we opt to use the term Heegaard
splitting to encompass a much larger class of splittings.

######  Remark 2.3.

Let us restrict attention here to sutured manifolds $(M,\gamma)$ without
spherical components of $R(\gamma)$. In [Juh06], Juhász introduces sutured
diagrams which are Heegaard diagrams for sutured manifolds $(M,\gamma)$.
Implicitly, these define Heegaard splittings for such sutured manifolds.
Conversely, given a Heegaard splitting $(A,B;S)$ of a sutured manifold
$(M,\gamma)$, a minimal choice of handle structure on $A$ and $B$ induces a
sutured diagram on $S$. We leave the details of this to the reader as we are
not concerning ourselves with sutured diagrams in this article.

Figure 4. Heegaard splitting of $(M,\gamma)$ into compression bodies $A$ and
$B$.

Every sutured manifold $M$ without toroidal sutures admits a Heegaard
splitting. The handle number of a Heegaard splitting $(M,S)=(A,B;S)$ of a
sutured manifold $M$ is $h(M,S)=h(A)+h(B)$. The (linear) handle number
$h(M,\gamma,0)$ of the sutured manifold $M$ is the minimum of $h(M,S)$ among
all linear Heegaard splittings of $(M,\gamma)$. Observe that this is the
minimum number of $0$–, $1$–, $2$–, and $3$–handles needed to construct $M$
from a collar of $R_{-}$.

In the case of a product sutured manifold $M=F\times[-1,1]$ where
$R_{\pm}(M)=F\times\pm 1$, the Heegaard surface $F\times 0$ gives a Heegaard
splitting of $M$ for which the compression bodies $A$ and $B$ are trivial.
Thus its linear handle number is zero. Indeed, the linear handle number of a
sutured manifold is $0$ exactly when the sutured manifold is a product.

While we often consider situations in which $M$ is connected, note that these
definitions all apply when $M$ is a disconnected sutured manifold. Indeed, a
Heegaard splitting of $M$ restricts to a Heegaard splitting on each connected
component and its handle number is the sum of the handle numbers of its
connected components. Furthermore, note that the Heegaard surface restricts to
a connected Heegaard surface for each connected component of $M$.

######  Remark 2.4.

Goda defines the (linear) handle number of a connected sutured manifold $M$ in
the situation where the surfaces $R_{+}$ and $R_{-}$ are non-empty and
homeomorphic (and have no sphere components). Then for any Heegaard splitting
$(A,B;S)$ of $M$, $h(A)$ is just a count of $1$–handles, and we have that
$h(A)=h(B)$. So Goda defines $h(M,\gamma)$ to be the minimum of $h(A)$ over
all Heegaard splittings [God06]. Our definition takes $h(M,\gamma,0)$ to be
twice that.

###  2.4. Circular Heegaard splittings

Let $(M,\gamma)$ be a sutured manifold, possibly with toroidal sutures. A
circular Heegaard splitting of $M$, or for simplicity just a Heegaard
splitting of $M$, is a decomposition of $M$ along a pair of disjoint properly
embedded oriented surfaces $F$ and $S$ so that

  * •

$F$ is a decomposing surface for $M$

  * •

the sutured manifold decomposition of $M$ along $F$ yields a sutured manifold
$M\backslash F$ without toroidal sutures, and

  * •

$S$ is a Heegaard surface for $M\backslash F$.

This last condition implies that $\partial S$ is contained in the union
$\gamma\cup\mathcal{N}(\partial F)$ of the sutures $\gamma$ and a collar
neighborhood of $\partial F$ in $\partial M$ so that $\partial S$ is the union
of the cores of the annular sutures of $M\backslash F$. Furthermore, as
$M\backslash F$ is potentially disconnected, the surface $S$ restricts to a
Heegaard surface on each component of $M\backslash F$. We may denote the
Heegaard splitting as the triple $(M,F,S)$ or the quadruple $(A,B;F,S)$ where
$(A,B;S)$ is a linear Heegaard splitting of $M\backslash F$. We say $S$ and
$F$ are the thick and thin surfaces of the splitting as they are respectively
the thick and thin boundaries of both $A$ and $B$, aside from $R_{+}$ and
$R_{-}$.

######  Remark 2.5.

Our definition of Heegaard splitting encompasses linear Heegaard splittings as
defined above as well as the splittings often called “generalized (linear)
Heegaard splittings” or “generalized circular Heegaard splittings”. Indeed a
linear Heegaard splitting is just a Heegaard splitting with $F=\emptyset$.

Observe that in a Heegaard splitting $(M,F,S)$, we have $[F,\partial
F]=[S,\partial S]\in H_{2}(M,\partial M)$. Furthermore, for each toroidal
suture $T$ of $\partial M$, each $[\partial F]$ and $[\partial S]$ must
restrict to the same non-zero class in $H_{1}(T)$. Let us say a class $\xi\in
H_{2}(M,\partial M)$ is regular (with respect to the sutured manifold
structure) if any surface $F$ representing $\xi$ meets each toroidal suture.

Note that if $(M,\gamma)$ has no toroidal sutures, then every class is
regular. However, if $(M,\gamma)$ has toroidal sutures, then a regular class
is necessarily non-zero. For the exterior of a knot in $S^{3}$ whose boundary
is a toroidal suture, then every non-zero class is regular. For the exterior
of a two-component link in $S^{3}$ whose boundary is a pair of toroidal
sutures, then every non-zero class is regular unless the linking number is
zero; in that case each component bounds a Seifert surface disjoint from the
other component.

The handle number of a (circular) Heegaard splitting $(M,F,S)=(A,B;F,S)$ is
$h(M,F,S)=h(A)+h(B)$. For a sutured manifold $(M,\gamma)$ and decomposing
surface $F$ that meets all toroidal sutures, the handle number $h(M,F)$ is the
minimum handle number $h(M,F,S)$ among Heegaard splittings of $(M,\gamma)$
with thin level $F$. (In other words, $h(M,F)$ is the linear handle number of
the sutured manifold $M\backslash F$ obtained by decomposing along $F$.) For a
sutured manifold $(M,\gamma)$ and regular class $\xi\in H_{2}(M,\partial M)$,
the handle number $h(M,\gamma,\xi)$ is the minimum handle number $h(M,F,S)$
among Heegaard splittings of $(M,\gamma)$ (or equivalently, the minimum handle
number $h(M,F)$ among decomposing surfaces) in which $F$ represents $\xi$,
i.e. in which $[F,\partial F]=\xi$. We say a surface $F$ representing a
regular class $\xi$ realizes $h(M,\gamma,\xi)$ if there is a surface $S$ in
$M$ such that $h(M,\gamma,\xi)=h(M,F,S)$.

Observe that $0$ is a regular class for a sutured manifold $(M,\gamma)$
without toroidal sutures. Hence the handle number for the trivial class is the
linear handle number $h(M,\gamma,0)$. This is in accordance with Remark 2.5
and Lemma 3.1 below.

###  2.5. Product decompositions and handle numbers

Consider a sutured manifold $(M,\gamma)$. A product disk is a properly
embedded disk whose boundary is a union of an arc in $R_{+}(\gamma)$, an arc
in $R_{-}(\gamma)$, and two spanning arcs of the annular sutures. A product
annulus is a properly embedded annulus with one boundary component in
$R_{+}(\gamma)$ and the other in $R_{-}(\gamma)$. A product decomposition is a
sutured manifold decomposition where the decomposing surface is a union of
product disks and product annuli.

The following lemma is well known in the theory of sutured manifold
decompositions. We include it without proof.

######  Lemma 2.6.

Suppose $E$ and $F$ are transversally intersecting decomposing surfaces for
$(M,\gamma)$ such that in any component of $A(\gamma)$ all arcs of $\partial
E$ and $\partial F$ are coherently oriented. Let $\delta$ be the collection of
product disks and product annuli induced from the double curve sum.

Then we have the following diagram of sutured manifold decompositions.

${{(M_{E},\gamma_{E})}}$${{(M,\gamma)}}$${{(M_{E\mathbin{\rotatebox[origin={c}]{-30.0}{$\asymp$}}F},\gamma_{E\mathbin{\rotatebox[origin={c}]{-30.0}{$\asymp$}}F})}}$${{(M^{\prime},\gamma^{\prime})}}$${{(M_{F},\gamma_{F})}}$$\scriptstyle{E}$$\scriptstyle{F}$$\scriptstyle{E\mathbin{\rotatebox[origin={c}]{-30.0}{$\scriptstyle\asymp$}}F}$$\scriptstyle{F\backslash
E}$$\scriptstyle{\delta}$$\scriptstyle{E\backslash F}$

∎

######  Remark 2.7.

The condition that all arcs of $\partial E$ and $\partial F$ are coherently
oriented in any component of $A(\gamma)$ will be called condition (C1) in
Section 3.3.

######  Lemma 2.8.

Suppose $(M,\gamma)$ has no toroidal sutures, and
$(M^{\prime},\gamma^{\prime})$ is obtained from $(M,\gamma)$ by a product
decomposition. Equivalently, suppose $(M,\gamma)$ is obtained from
$(M^{\prime},\gamma^{\prime})$ by gluing a pair of spanning rectangles in
$A(\gamma^{\prime})$ or a pair of components of $A(\gamma^{\prime})$. Then
$h(M^{\prime},\gamma^{\prime},0)\geq h(M,\gamma,0)$.

###### Proof.

We state the proof for a decomposition along a product disk. The case of a
product annulus carries through similarly.

Let $S^{\prime}$ be a Heegaard surface for $(M^{\prime},\gamma^{\prime})$ that
realizes $h(M^{\prime},\gamma^{\prime},0)$, splitting
$(M^{\prime},\gamma^{\prime})$ into compression bodies $A^{\prime}$ and
$B^{\prime}$. Let $\delta_{\pm}$ be two spanning rectangles in
$A(\gamma^{\prime})$ so that identifying them produces $(M,\gamma)$. Observe
that $S^{\prime}$ may be taken to intersect each of $\delta_{\pm}$ as the core
arc of the annular sutures does. The identification of $\delta_{\pm}$ may be
done to also identify these arcs so that $S^{\prime}$ becomes a surface $S$ in
the sutured manifold $(M,\gamma)$. Likewise, the identification of
$\delta_{\pm}$ restricts to an identification of the rectangles
$A^{\prime}\cap\delta_{\pm}$ of $A^{\prime}$ and also of the rectangles
$B^{\prime}\cap\delta_{\pm}$ of $B^{\prime}$. Consequently these
identifications yield sutured manifolds $A$ and $B$ with
$\partial_{+}A=S=\partial_{-}B$ that contain product disks, and the
decompositions along these product disks yield the compression bodies
$A^{\prime}$ and $B^{\prime}$. By [God92, Lemma 2.4], $A$ and $B$ are
compression bodies with $h(A)=h(A^{\prime})$ and $h(B)=h(B^{\prime})$. Hence
$S$ is a Heegaard surface for $(M,\gamma)$ with
$h(M,S)=h(A)+h(B)=h(A^{\prime})+h(B^{\prime})=h(M^{\prime},\gamma^{\prime},0)$.
Since $h(M,\gamma,0)\leq h(M,S)$, we have $h(M,\gamma,0)\leq
h(M^{\prime},\gamma^{\prime},0)$. ∎

###  2.6. Handle structures, handle counts, and handle indices

Given a handle structure on a manifold $X$, let $\mathcal{H}^{k}(X)$ be the
collection of its $k$–handles, and let $h^{k}(X)=|\mathcal{H}^{k}(X)|$ be the
number of $k$–handles.

The handle number of a sutured $3$–manifold $M$ with a handle structure is
$h(M)=h^{0}(M)+h^{1}(M)+h^{2}(M)+h^{3}(M)$. The handle index of a sutured
$3$–manifold $M$ with a handle structure is
$j(M)=(h^{1}(M)-h^{0}(M))+(h^{2}(M)-h^{3}(M))$.

Observe that for a minimal handle structure on a connected compression body
$A$ and a minimal handle structure on a connected dualized compression body
$B$ we have

$h^{0}(A)=\begin{cases}0&\mbox{ if }\partial_{-}A\neq\emptyset\\\ 1&\mbox{ if
}\partial_{-}A=\emptyset\end{cases}\qquad\mbox{ and }\qquad
h^{3}(B)=\begin{cases}0&\mbox{ if }\partial_{+}B\neq\emptyset\\\ 1&\mbox{ if
}\partial_{+}B=\emptyset\end{cases}$

so that

$h(A)=\begin{cases}j(A)&\mbox{ if }A\mbox{ is not a handlebody}\\\
j(A)+2&\mbox{ if }A\mbox{ is a handlebody}\end{cases}\qquad\mbox{ and }\qquad
h(B)=\begin{cases}j(B)&\mbox{ if }B\mbox{ is not a handlebody}\\\
j(B)+2&\mbox{ if }B\mbox{ is a handlebody}\end{cases}$

Let $(M,\gamma)$ be a connected sutured manifold without toroidal sutures.
Define $d=d(M,\gamma)$ to be $0$, $1$, or $2$ depending on whether neither,
just one, or both of $R_{+}$ and $R_{-}$ are empty.

Then for any linear Heegaard splitting of $(M,\gamma)$ realizing
$h(M,\gamma,0)$ with handle structure that is minimal on each compression
body, observe that $d=h^{0}(M)+h^{3}(M)$.

More generally, let $(M,F,S)=(F,S;A,B)$ be a circular Heegaard splitting of
the sutured manifold $(M,\gamma)$. Since
$h(M,F,S)=h(A)+h(B)=h^{0}(A)+h^{1}(A)+h^{2}(B)+h^{3}(B)$, the number
$d(M,F,S)=h^{0}(A)+h^{3}(B)$ counts the number of handlebodies in the
splitting. Then if $(M,F,S)$ is a splitting realizing $h(M,\gamma,\xi)$, we
may define $d(M,\gamma,\xi)=d(M,F,S)$. Observe that
$d(M,\gamma,0)=d(M,\gamma)$ as above while $d(M,\gamma,\xi)=0$ if $\xi\neq 0$.

##  3\. Some operations on Heegaard splittings

###  3.1. Amalgamations

Consider a circular Heegaard splitting $(M,F,S)$ of $M$ into a compression
body $A$ and dual compression body $B$. Suppose a collection of components
$A_{1}$ of $A$ and $B_{0}$ of $B$ meet along a collection of components
$F_{1}$ of $F$ (so that $F_{1}=\partial_{-}A_{1}\cap\partial_{+}B_{0}$). If
$\partial_{+}A_{1}\cap\partial_{-}B_{0}=\emptyset$ then we can amalgamate
$(M,F,S)$ along $F_{1}$ to produce a new circular Heegaard splitting
$(M^{\prime},F^{\prime},S^{\prime})$ with $F^{\prime}=F-F_{1}$.

Since we assume $\partial_{+}A_{1}\cap\partial_{-}B_{0}=\emptyset$, the
definition of amalgamation along a surface given in [Bak21, Definition 12]
extends to this case where $A_{1}$ and $B_{0}$ may be disconnected. (As noted
in Remark 2.5, our definition of circular Heegaard splitting encompasses
splittings that are often called generalized Heegaard splittings or
generalized circular splittings.) See also [Sch93, §2] and [Lac08, §3] for
other treatments of the amalgamation operation.

######  Lemma 3.1.

If $(M^{\prime},F^{\prime},S^{\prime})$ is an amalgamation of $(M,F,S)$, then
$h(M^{\prime},F^{\prime},S^{\prime})\leq h(M,F,S)$.

###### Proof.

From the definition of amalgamation (see [Bak21, Definition 12]), the initial
handle structure on $(M,F,S)$ extends to a handle structure on
$(M^{\prime},F^{\prime},S^{\prime})$ with the same number of handles of each
dimension. Hence we have $h(M^{\prime},F^{\prime},S^{\prime})\leq h(M,F,S)$.
(Note that equality does not necessarily hold since it is possible that
induced handle structure on $(M^{\prime},F^{\prime},S^{\prime})$ now allows
for some cancellations.) ∎

######  Remark 3.2.

If $(M^{\prime},F^{\prime},S^{\prime})$ is an amalgamation of $(M,F,S)$ along
the surface $F_{1}\subset F$, then observe that $[F^{\prime}]+[F_{1}]=[F]$.
Hence $[F^{\prime}]=[F]$ only if $[F_{1}]=0$.

###  3.2. Realization by homologically essential surfaces

A properly embedded surface $F$ in a $3$–manifold $M$ is homologically
essential if no non-empty subcollection of its components is null-homologous
in $H_{2}(M,\partial M)$.

######  Lemma 3.3.

Suppose $\xi$ is a regular class. If $\xi\neq 0$ and $F$ realizes
$h(M,\gamma,\xi)$, then there is a homologically essential subsurface
$\overline{F}\subset F$ that also realizes $h(M,\gamma,\xi)$.

If $\xi=0$, then $h(M,\gamma,0)$ is realized by a Heegaard splitting $(M,S)$.

###### Proof.

Suppose $F$ realizes $h(M,\gamma,\xi)$ with the Heegaard splitting $(M,F,S)$.
Further suppose $F$ contains a non-empty collection of components that is
null-homologous. Then $F$ contains a minimal such collection $F_{1}$, for
which no proper non-empty subcollection is null-homologous. In particular,
$F_{1}$ is the boundary of a submanifold of $M$. Then we have collections of
components $A_{1}$ of $A$ and $B_{0}$ of $B$ for which
$F_{1}=\partial_{-}A_{1}\cap\partial_{+}B_{0}$. They have their corresponding
collections of components $B_{1}$ of $B$ and $A_{0}$ of $A$ where
$\partial_{+}A_{i}=\partial_{-}B_{i}=S_{i}$ is a collection of components of
$S$ for $i=0,1$. Suppose $S_{0}\cap S_{1}\neq\emptyset$ so that $S_{0}$ and
$S_{1}$ share a component $S^{*}$. Then there are components $A^{*}$ of $A$
and $B^{*}$ of $B$ so that $S^{*}=\partial_{+}A^{*}=\partial_{-}B^{*}$ and
$F_{1}$ has a component in each $\partial_{-}A^{*}$ and $\partial_{+}B^{*}$.
Thus there is a path in $A^{*}\cup_{S^{*}}B^{*}$ from one side of $F_{1}$ to
the other side of $F_{1}$. However that is contrary to $F_{1}$ bounding a
submanifold of $M$. Thus $S_{0}\cap S_{1}=\emptyset$ and we may amalgamate
$(M,F,S)$ along $F_{1}$ to obtain the splitting
$(M^{\prime},F^{\prime},S^{\prime})$ where $F^{\prime}=F-F_{1}$. By Lemma 3.1,
$h(M^{\prime},F^{\prime},S^{\prime})\leq h(M,F,S)$. Since $F_{1}$ is null-
homologous, $[F^{\prime}]=[F]$ as noted in Remark 3.2. Since $F$ realizes
$h(M,\gamma,\xi)$, the previous inequality is an equality and $F^{\prime}$
realizes $h(M,\gamma,\xi)$ too. This may be repeated until we are left with a
Heegaard splitting $(M,\overline{F},\overline{S})$ that also realizes
$h(M,\gamma,\xi)$ where $\overline{F}\subset F$ is a homologically essential
subsurface, possibly the empty set. ∎

###  3.3. Conditioned splittings

Given a sutured manifold $(M,\gamma)$, a conditioned surface $F$ is a
decomposing surface of $(M,\gamma)$ such that

  1. (C1)

all arcs of $\partial F$ in any component of $A(\gamma)$ are coherently
oriented, and

  2. (C2)

no non-empty collection of simple closed curves $\partial F\cap R(\gamma)$ is
trivial in $H_{1}(\partial M,\gamma)$.

We further say $F$ is well-conditioned if

  1. (C3)

$F$ is homologically essential.

A Heegaard splitting $(M,F,S)$ of $(M,\gamma)$ is conditioned if $F$ is
conditioned, and it is well-conditioned if $F$ is well-conditioned.

######  Lemma 3.4.

Suppose $\xi$ is a regular class for $(M,\gamma)$. Then there is a surface $F$
realizing $h(M,\gamma,\xi)$ that satisfies condition (C1).

###### Proof.

Let $F_{0}$ be a decomposing surface for $(M,\gamma)$ that represents $\xi$.
Suppose two adjacent arcs of $F_{0}\cap A(\gamma)$ are anti-parallel and
joined by an arc $\alpha$ of the core suture. Then let $F_{1}$ be the surface
obtained by a boundary tubing of $F_{0}$ along $\alpha$ that is disjoint from
the rectangle of $A(\gamma)$ defined by alpha and these two arcs. Then $F_{1}$
has a boundary compressing disk $\delta_{\alpha}$ that crosses $\alpha$ once
and boundary compresses $F_{1}$ back into $F_{0}$. (Note that $F_{1}$ also
represents $\xi$.) In the result $(M_{1},\gamma_{1})$ of the sutured manifold
decomposition of $(M,\gamma)$ along $F_{1}$, $\delta_{\alpha}$ is a product
disk. Furthermore, the decomposition along $\delta_{\alpha}$ results in the
sutured manifold $(M_{0},\gamma_{0})$ obtained from decomposing $(M,\gamma)$
along $F_{0}$. Consequently, $(M_{1},\gamma_{1})$ can be regarded as being
obtained from $(M_{0},\gamma_{0})$ by regluing a product disk corresponding to
$\delta_{\alpha}$. See Figure 5. By Lemma 2.8, $h(M_{1},\gamma_{1},0)\leq
h(M_{0},\gamma_{0},0)$. In particular, there is a Heegaard surface $S_{1}$ of
$(M_{1},\gamma_{1})$ such that $h(M,F_{1},S_{1})\leq h(M,F_{0},S_{0})$ for any
Heegaard surface $S_{0}$ of $(M_{0},\gamma_{0})$.

Figure 5. Left: A boundary tubing of $F_{0}$ along $\alpha$ produces $F_{1}$
with its boundary compressing disk $\delta_{\alpha}$. Center: Upon decomposing
$(M,\gamma)$ along $F_{1}$ to obtain $(M_{1},\gamma_{1})$, the disk
$\delta_{\alpha}$ becomes a product disk. Right: The result of decomposing
$(M_{1},\gamma_{1})$ along the product disk $\delta_{\alpha}$ is homeomorphic
to $(M_{0},\gamma_{0})$ obtained by decomposing $(M,\gamma)$ along $F_{0}$.

Hence it follows that $h(M,\gamma,\xi)$ is realized by a surface $F$ whose
arcs of intersection with $A(\gamma)$ are coherently oriented in any component
of $A(\gamma)$. ∎

######  Lemma 3.5.

Let $F$ be a decomposing surface for a sutured manifold $(M,\gamma)$ that
meets every toroidal suture and satisfies property (C1). Let $R_{0}$ be a
component of $R=R(\gamma)$, let $F_{R_{0}}$ be the double curve sum of $F$
with a push-off of $R_{0}$, and let $F_{R_{0}}^{\prime}$ be the result of
discarding any components of $F_{R_{0}}$ that are $\partial$–parallel into
$R_{0}$. Then

  1. (1)

$F_{R_{0}}^{\prime}$ is homologous to $F$,

  2. (2)

$|F_{R_{0}}^{\prime}\cap R|\leq|F\cap R|$,

  3. (3)

$|F_{R_{0}}^{\prime}|\leq|F|$, and

  4. (4)

$h(M,F_{R_{0}}^{\prime})\leq h(M,F)$.

Furthermore,

  1. (2’)

$|F_{R_{0}}^{\prime}\cap R|<|F\cap R|$ if some component of $R_{0}\backslash
F$ gives the null-homology in $H_{1}(M,\gamma)$ of a non-empty collection of
simple closed curves of $\partial F\cap R_{0}$, and

  2. (3’)

$|F_{R_{0}}^{\prime}|<|F|$ if some component of $R_{0}\backslash F$ meets two
components of $F_{\pm}$ with consistent orientations.

###### Proof.

Fix a collar $C=R_{0}\times[0,1]$ in $M$ where $R_{0}\times\\{0\\}=R_{0}$ and
$\partial R_{0}\times[0,1]$ is contained in $A(\gamma)$ so that $F\cap C$ is a
collar in $F$ of $F\cap R_{0}$. In particular, $F\cap C$ may be identified
with $(\partial F\cap R_{0})\times[0,1]$. Then $F_{R_{0}}$ is the double curve
sum $R_{0}\times\\{1/2\\}\mathbin{\rotatebox[origin={c}]{-30.0}{$\asymp$}}F$.
Since $F$ satisfies property (C1), so does $F_{R_{0}}$. Furthermore, since
$R_{0}\times\\{1/2\\}$ is null-homologous in $H_{2}(M,\gamma)$, $F_{R_{0}}$ is
homologous to $F$. Then let $F_{R_{0}}^{\prime}$ be $F_{R_{0}}$ with any
component that is $\partial$–parallel into $R_{0}$ discarded, so
$F_{R_{0}}^{\prime}$ is homologous to $F$ too. Observe that $F\backslash
C=F_{R_{0}}\backslash C=F_{R_{0}}^{\prime}\backslash C$. See Figure 6.

Figure 6. The surfaces $F$, $F_{R_{0}}$, and $F_{R_{0}}^{\prime}$ (assuming
the lower-left piece of $F_{R_{0}}$ is not parallel into $R_{0}$).

Since $F\cap C$ is a collar of $\partial F\cap R_{0}$, $|F|=|F\backslash
C|=|F_{R_{0}}\backslash C|$. Since the components of $F_{R_{0}}$ that meet
$\partial F$ but not $F\backslash C$ are necessarily $\partial$–parallel into
$R_{0}$ since they are disjoint from $R_{0}\times\\{1\\}$. (Otherwise they
would meet $F\backslash C$). These are the components that are discarded to
form $F_{R_{0}}^{\prime}$. Consequently, $|F_{R_{0}}^{\prime}\cap R|\leq|F\cap
R|$. Now suppose there is a component $Q$ of $R_{0}\backslash F$ such that
$\partial Q$ is a union of components $\mathcal{C}$ of $\partial F$ (with
induced boundary orientation) and possibly some components of $\partial R_{0}$
(with any orientation). Then $[\mathcal{C}]=\partial[Q]$ in $H_{1}(\partial
M,\gamma)$. Furthermore, one component of the double curve sum $F_{R_{0}}$ is
formed from the union of $Q\times\\{1/2\\}$ and the annular collars
$\mathcal{C}\times[0,1/2]$ in $F$. This component is $\partial$–parallel to
$Q\subset R_{0}$ and is discarded when forming $F_{R_{0}}^{\prime}$. Thus
$\partial F_{R_{0}}^{\prime}$ is a proper subcollection of $\partial
F_{R_{0}}$, and hence $|F_{R_{0}}^{\prime}\cap R|<|F\cap R|$.

As $F_{R_{0}}^{\prime}$ is the union of the components of $F_{R_{0}}$ that
meet $F\backslash C$, it has at most as many components as $F$. Hence
$|F_{R_{0}}^{\prime}|\leq|F|$. However, further observe that if some component
$Q$ of $R_{0}\backslash F$ meets two components of $F_{\pm}$ with consistent
orientations (that is, if say $R_{0}\subset R_{-}$, then this component $Q$
should meet two components of $F_{-}$), then in the double curve sum
$Q\times\\{1/2\\}$ is joined to $(Q\cap F)\times[1/2,1]$ which then joins to
the corresponding components of $F\backslash C$. Hence $F_{R_{0}}^{\prime}$
will have fewer components than $F_{R_{0}}^{\prime}\backslash C$. Since
$|F|=|F\backslash C|$ and $F\backslash C=F_{R_{0}}^{\prime}\backslash C$, we
have that $|F_{R_{0}}^{\prime}|<|F|$.

Finally, to show that $h(M,F_{R_{0}}^{\prime})\leq h(M,F)$, observe that
$h(M,F_{R_{0}})=h(M,F_{R_{0}}^{\prime})$ immediately follows from the
definition of $F_{R_{0}}^{\prime}$. So we aim to show that $h(M,F_{R_{0}})\leq
h(M,F)$. Let $(M_{F_{R_{0}}},\gamma_{F_{R_{0}}})$ be the sutured manifold
decomposition of $(M,\gamma)$ along $F_{R_{0}}$. Let
$R_{1/2}=R_{0}\times\\{1/2\\}$. Lemma 2.6 implies we have the following
diagram:

${{(M_{R_{1/2}},\gamma_{R_{1/2}})}}$${{(M,\gamma)}}$${{(M_{F_{R_{0}}},\gamma_{F_{R_{0}}})}}$${{(M^{\prime},\gamma^{\prime})}}$${{(M_{F},\gamma_{F})}}$$\scriptstyle{R_{1/2}}$$\scriptstyle{F}$$\scriptstyle{F_{R_{0}}}$$\scriptstyle{F\backslash
R_{1/2}}$$\scriptstyle{\delta}$$\scriptstyle{R_{1/2}\backslash F}$

where $\delta$ is a set of product disks and annuli in
$(M_{F_{R_{0}}},\gamma_{F_{R_{0}}})$ arising from the double curve sum of $F$
and $R_{1/2}$ that produces $F_{R_{0}}$. We observe that:

  * •

$(M_{R_{1/2}},\gamma_{R_{1/2}})$ is homeomorphic to
$(M,\gamma)\cup(R_{0}\times[0,1/2],\gamma\cap(R_{0}\times[0,1/2]))$,

  * •

$F\backslash R_{1/2}\cong F\cup(\partial F\cap R_{0})\times[0,1/2]$, and

  * •

$R_{1/2}\backslash F$ is parallel to $R_{0}\backslash F$ in
$(M_{F},\gamma_{F})$.

Hence $(M^{\prime},\gamma^{\prime})$ is the union of $(M_{F},\gamma_{F})$ and
a product sutured manifold. So
$h(M^{\prime},\gamma^{\prime},0)=h(M_{F},\gamma_{F},0)=h(M,F)$. Thus we have

$h(M,F)=h(M_{F},\gamma_{F},0)=h(M^{\prime},\gamma^{\prime},0)\geq
h(M_{F_{R_{0}}},\gamma_{F_{R_{0}}},0)=h(M,F_{R_{0}})$

where the inequality follows from Lemma 2.8. ∎

######  Theorem 3.6.

For any sutured manifold $(M,\gamma)$ and regular class $\xi\in
H_{2}(M,\partial M)$ there is a well-conditioned Heegaard splitting that
realizes $h(M,\gamma,\xi)$.

###### Proof.

By Lemma 3.4, there is a surface realizing $h(M,\gamma,\xi)$ that satisfies
condition (C1). Assume that $F_{0}$ has been chosen among all such surfaces
$F$ to minimize the complexity $(|F|,|F\cap R|)$ lexicographically.

Suppose that there is a non-empty collection of simple closed curves $\partial
F_{0}\cap R(\gamma)$ is trivial in $H_{1}(\partial M,\gamma)$. Then there is a
component $R_{0}$ of $R$ that contains such a collection. In particular, there
is an oriented collection $\mathcal{C}=C_{1}\cup\dots\cup C_{n}$ of components
of $\partial F_{0}$ in $R_{0}$ so that $\mathcal{C}$ bounds (with perhaps some
components of $\partial R_{0}$) a connected subsurface $Q$ of $R_{0}$ that
contains no other such collection. If the interior of $Q$ is disjoint from
$\partial F_{0}$, then Lemma 3.5 produces a surface with smaller complexity.
Hence $\partial F_{0}$ must intersect the interior of $Q$. In particular there
is a component $C_{0}$ of $\partial F_{0}$ (possibly an arc) in the interior
of $Q$ and an arc $\alpha$ embedded in $Q$ running from $\mathcal{C}$ to the
same side of $C_{0}$ that is otherwise disjoint from $\partial F_{0}$. (Due to
condition (C1), all arcs of $Q\cap\partial F_{0}$ must meet any component of
$\partial R_{0}$ in $\partial Q$ coherently. So if there is an arc
$\alpha^{\prime}$ embedded in $Q$ running from $\mathcal{C}$ to an arc of
$Q\cap\partial F_{0}$, then there is an arc $\alpha$ as described.) A boundary
tubing of $F_{0}$ along $\alpha$ produces a surface $F_{\alpha}$ so that
$M\backslash F_{\alpha}$ decomposed along a product disk $\delta_{\alpha}$
(the boundary compression of the tube along $\alpha$) gives the same as
$M\backslash F_{0}$, see Figure 7. So by Lemma 2.8 $F_{\alpha}$ must also
realize the handle number. But $F_{\alpha}$ meets $R$ in fewer components,
contrary to assumption. Thus $F_{0}$ must satisfy condition (C2).

Figure 7. A boundary tubing of a thin surface $F_{0}$ (left) along an arc
$\alpha$ produces a surface $F_{\alpha}$ (right). The induced boundary
compressing disk $\delta_{\alpha}$ of $F_{\alpha}$ is a product disk for the
decomposition along $F_{\alpha}$.

Since any collection of components of a conditioned surface continues to be
conditioned, Lemma 3.3 shows that we may discard any homologically inessential
components of $F_{0}$ to obtain a surface $\overline{F}_{0}$ satisfying
condition (C3) that also realizes the handle number.

In particular, $\overline{F}_{0}$ is well-conditioned and we have a well-
conditioned splitting $(M,\overline{F}_{0},\overline{S}_{0})$ that realizes
$h(M,\gamma,\xi)$. ∎

###  3.4. Tubings of splittings

Let $(M,F,S)$ be a Heegaard splitting of $(M,\gamma)$ with a non-trivial
compression body $A_{i}$ containing a $1$–handle ${\mathcal{H}}^{1}_{i}$ whose
feet are in $F$. Let $B_{j}$ and $B_{k}$ be the dual compression bodies that
meet the feet of ${\mathcal{H}}^{1}_{i}$. (Note that we may have
$B_{j}=B_{k}$.) Let $A_{j}$ and $A_{k}$ be the corresponding compression
bodies where $\partial_{+}A_{j}=\partial_{-}B_{j}=S_{j}$ and
$\partial_{+}A_{k}=\partial_{-}B_{k}=S_{k}$ are each components of $S$. Form
the compression body $A_{i}^{\prime}=A_{i}\backslash{\mathcal{H}}^{1}_{i}$. In
$B_{j}\cup B_{k}\cup{\mathcal{H}}^{1}_{i}$, form a $1$–handle
${\mathcal{H}}^{1}_{jk}$ with feet in $S_{j}\cup S_{k}$ from a (narrower)
regular neighborhood of the core of ${\mathcal{H}}^{1}_{i}$ and two vertical
arcs in $B_{j}$ and $B_{k}$ joining the endpoints of the core to $S_{j}\cup
S_{k}$. Now $B_{jk}^{\prime}=(B_{j}\cup
B_{k}\cup{\mathcal{H}}^{1}_{i})\backslash{\mathcal{H}}^{1}_{jk}$ is a dual
compression body and $A_{jk}^{\prime}=A_{j}\cup
A_{k}\cup{\mathcal{H}}^{1}_{jk}$ is a compression body that meet along the new
thick surface $S^{\prime}_{jk}$ that is the tubing of $S_{j}\cup S_{k}$ along
$\partial{\mathcal{H}}^{1}_{jk}$. Also observe that the compression body
$A_{i}^{\prime}=A_{i}\backslash{\mathcal{H}}^{1}_{i}$ and the dual compression
body $B_{jk}^{\prime}=(B_{j}\cup
B_{k}\cup{\mathcal{H}}^{1}_{i})\backslash{\mathcal{H}}^{1}_{jk}$ meet along
the thin surface $F^{\prime}_{jk}$ that is the tubing of $\partial_{-}A_{j}$
and $\partial_{-}A_{k}$ along $\partial{\mathcal{H}}^{1}_{jk}$. This forms a
new Heegaard splitting $(M,F^{\prime},S^{\prime})$ that we say is the tubing
of $(M,F,S)$ along the $1$–handle ${\mathcal{H}}^{1}_{i}$. See Figure 8.

Figure 8. Left: A Heegaard splitting $(M,F,S)$ with a $1$–handle
${\mathcal{H}}^{1}_{i}$ in $A_{i}$ whose feet are in $F$. Right: The new
Heegaard splitting $(M,F^{\prime},S^{\prime})$ is the tubing of $(M,F,S)$
along the $1$–handle ${\mathcal{H}}^{1}_{i}$.

######  Lemma 3.7.

If $(M,F^{\prime},S^{\prime})$ is a tubing of $(M,F,S)$, then
$[F^{\prime}]=[F]$, $|F^{\prime}|\leq|F|$, and $h(M,F^{\prime},S^{\prime})\leq
h(M,F,S)$. In particular, if the tubing occurs along a handle with feet in
distinct components of $F$, then $|F^{\prime}|=|F|-1$.

###### Proof.

From the definition of tubing along a handle, the initial handle structure of
$(M,F,S)$ extends to a handle structure on
$(M^{\prime},F^{\prime},S^{\prime})$ with the same number of handles of each
dimension. Hence $h(M,F^{\prime},S^{\prime})\leq h(M,F,S)$. Also, from the
definition, if the tubing is done along a handle with feet in distinct
components of $F$, then it follows that $|F^{\prime}|=|F|-1$. ∎

######  Remark 3.8.

Observe that the operation of tubing a splitting preserves the property of
being well-conditioned. That is, if $(M,F^{\prime},S^{\prime})$ is a tubing of
a well-conditioned splitting $(M,F,S)$, then $(M,F^{\prime},S^{\prime})$ is
well-conditioned too.

##  4\. Restructuring sutures

To get around the issue of non-zero classes that may fail to be regular for
link exteriors where the boundary tori are all toroidal sutures, we may
instead choose the boundary tori to all belong to $R_{\pm}$. This latter
sutured manifold structure allows the circular handle number to be defined for
all non-zero classes by ensuring they are all regular. Moreover, for regular
classes of the former “all toroidal suture” sutured manifold structure, the
resulting circular handle number is the same as with this latter “all
$R_{\pm}$” structure. More generally, we have the following.

######  Theorem 4.1.

Consider sutured manifolds structures $(M,\gamma)$ and $(M,\gamma^{*})$ which
agree on $\partial M\backslash T(\gamma)$ and for which
$T(\gamma^{*})=\emptyset$.

If $\xi\in H_{2}(M,\partial M)$ is a regular class for $(M,\gamma)$, then
either

  * •

$h(M,\gamma,\xi)=h(M,\gamma^{*},\xi)$,

  * •

there is a component of $T(\gamma)$ in which $\partial\xi$ and $A(\gamma^{*})$
have the same slope, or

  * •

there is a component of $T(\gamma)$ in which some component of
$R_{\pm}(\gamma^{*})$ is a disk.

In particular, if $T(\gamma)\subset R_{\pm}(\gamma^{*})$ then
$h(M,\gamma,\xi)=h(M,\gamma^{*},\xi)$ for every regular class of $(M,\gamma)$.

###### Proof.

Let us assume the last conclusion of the theorem does not occur; that is, in
no component of $T(\gamma)$ is some component of $R_{\pm}(\gamma^{*})$ a disk.
Then this theorem is primarily a consequence of the following observation. Let
${\mathcal{T}}$ be a collection of torus boundary components of a $3$–manifold
$M$. Let $F$ be a properly embedded, oriented surface in $M$ whose boundary
meets each component of ${\mathcal{T}}$ in a non-empty collection of
coherently oriented essential simple closed curves. Then if $(M,\gamma)$ and
$(M,\gamma^{*})$ are two sutured manifold structures on $M$ that differ only
on ${\mathcal{T}}$ such that $F$ is a decomposing surface for each, the
sutured manifolds that result from their decompositions along $F$ are
homeomorphic unless one has annular sutures on a component of ${\mathcal{T}}$
of the same slope as the curves of $\partial F$.

Now, with the hypotheses of the theorem, suppose $\xi\in H_{2}(M,\partial M)$
is a regular class for $(M,\gamma)$. Then for every torus $T$ in $T(\gamma)$,
the restriction of $\partial\xi$ to $H_{1}(T)$ is not $0$. So assume the
following:

> ($\dagger$) The slopes of $\partial\xi$ and $A(\gamma^{*})$ are distinct in
> each component of $T(\gamma)$.

Hence if $F$ is a decomposing surface for $(M,\gamma)$ representing $\xi$, it
meets each component of $T(\gamma)$ in a non-empty collection of coherently
oriented simple closed curves. By the above observation with our assumption
($\dagger$), the results of decomposing each $(M,\gamma)$ and $(M,\gamma^{*})$
along $F$ are homeomorphic to the same sutured manifold
$(M^{\prime},\gamma^{\prime})$. Then for any splitting $(M,F,S)$ of
$(M,\gamma)$ realizing $h(M,\gamma,\xi)$ there is a splitting $(M,F,S^{*})$ of
$(M,\gamma^{*})$ with $h(M,F,S^{*})=h(M,F,S)=h(M,\gamma,\xi)$. Hence
$h(M,\gamma^{*},\xi)\leq h(M,\gamma,\xi)$.

On the other hand, suppose $(M,F^{*},S^{*})$ is a splitting of
$(M,\gamma^{*})$ realizing $h(M,\gamma^{*},\xi)$. By Theorem 3.6, we may
assume $(M,F^{*},S^{*})$ is well-conditioned. In particular, this means
$\partial F^{*}$ is coherently oriented on each component of $T(\gamma)$.
Then, again by the above observation that begins this proof with our
assumption ($\dagger$), the results of decomposing each $(M,\gamma)$ and
$(M,\gamma^{*})$ along $F^{*}$ are homeomorphic. Hence there is a splitting
$(M,F^{*},S)$ of $(M,\gamma)$ with
$h(M,F^{*},S)=h(M,F^{*},S^{*})=h(M,\gamma^{*},\xi)$, and so
$h((M,\gamma,\xi)\leq h(M,\gamma^{*},\xi)$.

Together this implies that, under assumption ($\dagger$), we have
$h((M,\gamma,\xi)=h(M,\gamma^{*},\xi)$. Furthermore, if $T(\gamma)\subset
R_{\pm}(\gamma^{*})$, then $A(\gamma^{*})\cap T(\gamma)=\emptyset$ so that
assumption ($\dagger$) holds for all regular classes $\xi$ of $(M,\gamma)$. ∎

A similar proof gives the following.

######  Corollary 4.2.

Consider sutured manifolds structures $(M,\gamma_{1})$ and $(M,\gamma_{2})$
which only disagree on a collection of tori of $\partial M$ in which
$A(\gamma_{i})$ are essential annuli.

If $\xi\in H_{2}(M,\partial M)$ is a regular class for $(M,\gamma_{1})$ and
$(M,\gamma_{2})$, then either

  * •

$h(M,\gamma_{1},\xi)=h(M,\gamma_{2},\xi)$ or

  * •

there is a torus component of $\partial M$ in which $\partial\xi$ and
$A(\gamma_{i})$ have the same slope for either $i=1,2$. ∎

##  5\. An upper bound for sutured manifolds without toroidal sutures

###  5.1. Decompositions of compression bodies

Given a connected compact oriented surface $\Sigma$, the product
$P=\Sigma\times I$ is a trivial compression body which has the structure of a
product sutured manifold $(P,\gamma)$ where the sutures
$\gamma=\partial\Sigma\times\\{0\\}$ have annular neighborhoods
$A(\gamma)=\partial\Sigma\times I$. Here we take $I=[-1,1]$. Furthermore we
have the oriented surfaces $R_{\pm}=\Sigma\times\\{\pm 1\\}$ for which the
normal orientation is the $I$ direction. Being a product sutured manifold,
$(P,\gamma)$ is taut, e.g. [Sch89, Definition 2.2].

######  Theorem 5.1.

Let $\Sigma$ be a connected compact oriented surface other than a sphere. Let
$P=\Sigma\times I$ be the induced product sutured manifold. Let $F_{0}$ be a
conditioned surface in $P$. Then there is a conditioned surface $F$ in $P$
where

  1. (1)

$\partial F\backslash A(\gamma)=\partial F_{0}\backslash A(\gamma)$,

  2. (2)

$\partial F$ is homologous to $[\partial F_{0}]+m[\gamma]$ in $H_{1}(\partial
P)$ for some non-negative integer $m$,

  3. (3)

$[F,\partial F]=[F_{0},\partial F_{0}]$ in $H_{2}(P,\partial P)$,

  4. (4)

$F$ is taut, and

  5. (5)

the sutured manifold decomposition of $(P,\gamma)$ along $F$ is again a
product.

In particular, if $\Sigma$ is closed so that $A(\gamma)=\emptyset$, then
$\partial F=\partial F_{0}$.

###### Proof.

As $P=\Sigma\times I$ and $\Sigma\not\cong S^{2}$, it is irreducible and
$\Sigma\times\\{0\\}$ is a taut surface. Setting $y=[F_{0},\partial F_{0}]\in
H_{2}(P,\partial P)$ so that $\partial y=[\partial F_{0}]$, [Sch89, Theorem
2.5] implies there exists a taut surface $F$ in $P$ so that $\partial
F\backslash A(\gamma)=\partial F_{0}\backslash A(\gamma)$ and $[F,\partial
F]=y+m[\Sigma\times\\{0\\},\partial\Sigma\times\\{0\\}]$ in $H_{2}(P,\partial
P)$ for some non-negative integer $m$. In particular, this implies $[\partial
F]=[\partial F_{0}]+m[\gamma]$ in $H_{1}(\partial P)$ since
$\gamma=\partial\Sigma\times\\{0\\}$. We may further assume that any
oppositely oriented components of $\partial F$ in $A(\gamma)$ or $T(\gamma)$
have been capped off with annuli so that $F$ is a conditioned surface and any
closed separating components have been discarded (as done in the proof of
[Sch89, Theorem 2.6]).

Let $(P^{\prime},\gamma^{\prime})$ be the sutured manifold obtained from
decomposing $(P,\gamma)$ along $F$. Since $\Sigma\times\\{\pm 1\\}$ is a pair
of similarly oriented parallel copies of $\Sigma\times\\{0\\}$, [Sch89,
Theorem 2.5(iii)] implies that the double curve sum
$F\mathbin{\rotatebox[origin={c}]{-30.0}{$\asymp$}}\Sigma\times\\{\pm 1\\}$ is
taut in $P$, and hence also taut in $P^{\prime}=P\backslash F$ (again, see
also the proof of [Sch89, Theorem 2.6]). In particular, $R(\gamma^{\prime})$
is taut in $P^{\prime}$. Next, note that any sphere in $P^{\prime}$ bounds a
$3$–ball in $P$ (because $P$ is irreducible) that is disjoint from $F$ since
$F$ has no closed separating components; thus $P^{\prime}$ is irreducible.
Therefore, by [Sch89, Definition 2.2], this shows that the sutured manifold
$(P^{\prime},\gamma^{\prime})$ is a taut sutured manifold: we have that both
$P^{\prime}$ is irreducible and $R(\gamma^{\prime})$ is taut. Thus the sutured
manifold decomposition
$(P,\gamma)\overset{F}{\leadsto}(P^{\prime},\gamma^{\prime})$ is taut. Now
[Gab86, Lemma 2.4] (see also [Gab83, §4]) implies that the resulting sutured
manifold is also a product manifold. Thus $F$ satisfies the conclusion. ∎

######  Lemma 5.2.

Let $C$ be a connected compression body with handle number $h(C)$. For a
handle structure on $C$ realizing $h(C)$, let ${\mathcal{D}}$ be a collection
of coherently oriented copies of meridional disks that are co-cores of
$1$–handles. Let ${\mathcal{A}}$ be a collection of disks and annuli that are
vertical with respect to this handle structure. Then the decomposition of $C$
along ${\mathcal{D}}\cup{\mathcal{A}}$ is a compression body $C^{\prime}$ with
handle number $h(C^{\prime})=h(C)$ if $\partial_{-}C\neq\emptyset$ and
$h(C^{\prime})=h(C)-2$ if $C$ is a handlebody. In either case, the handle
index is preserved: $j(C^{\prime})=j(C)$.

###### Proof.

First decompose $C$ along ${\mathcal{A}}$. Since ${\mathcal{A}}$ is a
collection of vertical surfaces, $C\backslash{\mathcal{A}}$ is again a
compression body. Moreover, viewing $C\backslash{\mathcal{A}}$ as a
submanifold of $C$, it inherits the handle structure of $C$. In particular,
$h(C\backslash{\mathcal{A}})=h(C)$ and ${\mathcal{D}}$ continues to be a
collection of coherently oriented copies of meridional disks that are co-cores
of $1$-handles of this inherited handle structure. So we continue the argument
with the compression body $C\backslash{\mathcal{A}}$. Hence we may as well
proceed assuming ${\mathcal{A}}=\emptyset$.

One observes that the region between a pair of parallel and adjacent disks in
${\mathcal{D}}$ yields a product ball after the decomposition. Since products
may be components of compression bodies and they contribute nothing towards
the handle number, we further reduce to assuming ${\mathcal{D}}$ has no
parallel disks. Say $d=|{\mathcal{D}}|$. Then $C$ has a handle structure in
which the disks of ${\mathcal{D}}$ are co-cores of some of the handles.
Decomposing $C$ along ${\mathcal{D}}$ therefore cuts these handles open.
Forgetting about any sutured manifold structure,
$C^{\prime}=C\backslash{\mathcal{D}}$ is a compression body with handle number
$h(C)-d$. Each disk of ${\mathcal{D}}$ leaves two impressions in $\partial
C^{\prime}$. So, keeping track of the sutures induced from the decomposition
of $C$ along ${\mathcal{D}}$, we have a sutured manifold structure on
$C^{\prime}$ where $R_{-}(C^{\prime})$ is the disjoint union of $R_{-}(C)$ and
a collection of $d$ disks in $\partial C^{\prime}$, one for each disk in
${\mathcal{D}}$. $R_{+}(C^{\prime})$ is then just $\partial
C^{\prime}-\mathcal{N}(R_{-}(C^{\prime}))$; indeed $R_{+}(C^{\prime})$ is
obtained by adding a copy of each disk in ${\mathcal{D}}$, with the
appropriate orientation, to what is left of $R_{+}(C)$. Thus, as a sutured
manifold, $C^{\prime}$ is a compression body obtained by joining a thickening
of these $d$ disks of to a thickening of $R_{-}(C)$ each with a $1$-handle, if
$R_{-}(C)\neq\emptyset$. Thus $h(C^{\prime})=h(C)$. Figure 9(Left) illustrates
this situation.

Figure 9. Left: A compression body cut along a collection ${\mathcal{D}}$.
Right: A handlebody cut along another collection ${\mathcal{D}}$

If $R_{-}(C)=\emptyset$ (so that $C$ is a handlebody) then $R_{-}(C^{\prime})$
is a collection of $d$ disks in $\partial C^{\prime}$, one for each disk in
${\mathcal{D}}$. Then, as a sutured manifold, $C^{\prime}$ is obtained by
first joining a thickening of these $d$ disks to the $0$–handle of $C$ with
$d$ $1$–handles and then adding the remaining $h(C)-d-1$ $1$–handles that were
disjoint from the disks of ${\mathcal{D}}$. Now, with a choice of one of the
$1$–handles from a thickened disk of $R_{-}(C^{\prime})$ to the $0$–handle,
the other $1$–handles may be slid off the $0$–handle and down this $1$–handle
to the thickened disk. Finally, the $0$–handle and this $1$–handle may be
cancelled. Thus $h(C^{\prime})=h(C)-2$. Figure 9(Right) illustrates this
situation. ∎

The following theorem is a generalization of [God92, Lemma 2.4] which says
that a sutured manifold is a compression body if and only if it is obtained
after decomposing a compression body along a product disk, and that the handle
number is preserved under such decomposition.

######  Theorem 5.3.

Let $C$ be a connected compression body with sutures $\gamma$ and handle
number $h(C)$. Let $F_{0}$ be a conditioned surface. Then there is a
conditioned surface $F$ in $C$ such that $\partial F\backslash
A(\gamma)=\partial F_{0}\backslash A(\gamma)$ and $[F,\partial
F]=[F_{0},\partial F_{0}]+m[\partial_{+}C,\partial(\partial_{+}C)]$ in
$H_{2}(C,\partial C)$ for some integer $m>0$ with the following property. The
decomposition of $C$ along $F$ is a compression body $C^{\prime}$ with handle
number $h(C^{\prime})=h(C)$ if $\partial_{-}C\neq\emptyset$ and
$h(C^{\prime})=h(C)-2$ if $C$ is a handlebody. In either case the handle index
is preserved: $j(C^{\prime})=j(C)$.

###### Proof.

A spine of a compression body $C$ is the union of $\partial_{-}C$ and an
embedded graph $\tau$ so that
$C\backslash(\partial_{-}C\cup\tau)\cong\partial_{+}C\times[0,1]$. Choose a
spine $\partial_{-}C\cup\tau$ of $C$ induced by a handle structure realizing
$h(C)$, and consider a thin closed neighborhood
$N=\partial_{-}C\times[-1,-1+\epsilon]\cup N_{\tau}$ where $N_{\tau}$ is a
closed regular neighborhood of $\tau$. Note that $N$ is a compression body
homeomorphic to $C$ and $C\backslash N$ is the product sutured manifold
$P=\partial_{+}C\times I$. Let $\Sigma$ be the surface
$\partial_{+}N=\partial_{-}P$ so that $C=N\cup_{\Sigma}P$.

Observe that $F_{0}$ may be isotoped in $C$ to meet this spine so that
$F_{0}\cap\partial_{-}C$ is properly embedded collection of arcs and loops in
$\partial_{-}C$ and $F_{0}$ intersects $\tau$ in a discrete set of interior
points. Thus we may assume $F_{0}$ intersects $N$ in a collection of
meridional disks, vertical disks, and vertical annuli. Adjust $F_{0}$ by
tubing along subarcs of $\tau$ if necessary to ensure that within any
parallelism class these meridional disks are coherently oriented, and update
$F_{0}$ to be the resulting surface. This preserves that $F_{0}$ is
conditioned. Since $F_{0}$ is conditioned, both $F_{0}\cap N$ and $F_{0}\cap
P$ are conditioned surfaces in $N$ and $P$. Set $F_{N}=F_{0}\cap N$. Next
apply Theorem 5.1 to the surface $F_{0}\backslash N=F_{0}\cap P$ in the
product $P$ to obtain a conditioned surface $F_{P}$. Then together these form
the conditioned surface $F=F_{P}\cup F_{N}$ which is properly embedded in $C$.

We claim that $F$ is now our desired surface. Since $F\cap N=F_{N}=F_{0}\cap
N$ while $F\cap P=F_{P}$ is obtained from Theorem 5.1, we constructed $F$ so
that $\partial F\backslash A(\gamma)=\partial F_{0}\backslash A(\gamma)$ and
and $[F,\partial F]=[F_{0},\partial
F_{0}]+m[\partial_{+}C,\partial(\partial_{+}C)]$ in $H_{2}(C,\partial C)$ for
some integer $m>0$. Furthermore, it gives the desired decomposition.

Indeed, the result $C^{\prime}$ of the decomposition of $C$ along $F$ is the
union of the compression body $N^{\prime}=N\backslash F_{N}$ from Lemma 5.2
and the product $P^{\prime}=P\backslash F_{P}$ from Theorem 5.1 along the
impressions of surface $\Sigma^{\prime}=\Sigma\backslash F$ in $\partial
N^{\prime}$ and $\partial P^{\prime}$. More specifically, since $F$
transversally intersects $\Sigma$, the decomposition of $C=N\cup_{\Sigma}P$ as
a sutured manifold along $F$ may be viewed as follows. Figure 10 gives a
schematic overview.

  * •

First we decompose $C$ along $\Sigma$ into the product $P$ and compression
body $N$. Under this decomposition, $\Sigma$ leaves impressions
$\Sigma_{+}=R_{+}(N)$ and $\Sigma_{-}=R_{-}(P)$ and splits $F$ into
$F_{N}\subset N$ and $F_{P}\subset P$. Note that $R_{-}(N)=R_{-}(C)$ and
$R_{+}(P)=R_{+}(C)$.

  * •

Next we decompose $N$ along $F_{N}$ to obtain the compression body
$N^{\prime}$ and $P$ along $F_{P}$ to obtain the product $P^{\prime}$.
Identifying $\Sigma^{\prime}$ as $\Sigma\backslash F$, we observe that
$R_{+}(N^{\prime})=\Sigma^{\prime}_{+}\cup(F_{N})_{+}$ and
$R_{-}(P^{\prime})=\Sigma^{\prime}_{-}\cup(F_{P})_{-}$. Furthermore
$R_{-}(N^{\prime})=(R_{-}(N)\backslash F_{N})\cup(F_{N})_{-}$ and
$R_{+}(P^{\prime})=(R_{+}(P)\backslash F_{P})\cup(F_{P})_{+}$.

  * •

Finally we glue $P^{\prime}$ to $N^{\prime}$, identifying
$\Sigma^{\prime}_{+}\subset R_{+}(N^{\prime})$ with
$\Sigma^{\prime}_{-}\subset R_{-}(P^{\prime})$. This forms a sutured manifold
$C^{\prime}$ and a properly embedded surface $\Sigma^{\prime}$ arising from
the identification of $\Sigma^{\prime}_{+}$ with $\Sigma^{\prime}_{-}$. An
interval of $\partial\Sigma^{\prime}$ belongs to the sutures of $C^{\prime}$
only if it is in both or neither of $\partial\Sigma^{\prime}_{+}$ and
$\partial\Sigma^{\prime}_{-}$; see Figure 11. (By construction, every point of
$\partial\Sigma^{\prime}$ is in at least one of $\partial\Sigma^{\prime}_{+}$
or $\partial\Sigma^{\prime}_{-}$.) Thus we have
$R_{-}(C^{\prime})=R_{-}(N^{\prime})\cup(F_{P})_{-}$ and
$R_{+}(C^{\prime})=R_{+}(P^{\prime})\cup(F_{N})_{+}$.

  * •

Therefore, because $F_{-}=(F_{N})_{-}\cup(F_{P})_{-}$, and
$F_{+}=(F_{N})_{+}\cup(F_{P})_{+}$, we have $R_{-}(N)\backslash
F_{N}=R_{-}(C)\backslash F$ and $R_{+}(P)\backslash F_{P}=R_{+}(C)\backslash
F$. Then it follows that

$R_{-}(C^{\prime})=R_{-}(N^{\prime})\cup(F_{P})_{-}=(R_{-}(N)\backslash
F_{N})\cup(F_{N})_{-}\cup(F_{P})_{-}=(R_{-}(C)\backslash F)\cup F_{-}$

and

$R_{+}(C^{\prime})=R_{+}(P^{\prime})\cup(F_{N})_{+}=(R_{+}(P)\backslash
F_{P})\cup(F_{N})_{+}\cup(F_{P})_{+}=(R_{+}(C)\backslash F)\cup F_{+}.$

So we see that $C^{\prime}$ is the sutured manifold obtained from the
decomposition of $C$ along $F$.

Figure 10. A schematic of how decompositions and gluings of sutured manifolds
behave near the surfaces $\Sigma$ and $F$ in $C$. In particular, the
decomposition along $F$ may be recovered by a decomposition along $\Sigma$,
followed by a decomposition along $F_{P}\cup F_{N}$, and gluing the resulting
sutured manifolds together along $\Sigma^{\prime}_{+}$ and
$\Sigma^{\prime}_{-}$.  Figure 11. A schematic of how sutures behave under
gluing a sutured manifold along subsurfaces $\Sigma^{\prime}_{-}\subset R_{-}$
and $\Sigma^{\prime}_{+}\subset R_{+}$. Figure 12. The compression body
$C^{\prime}$ is obtained by attaching the product over $\Sigma^{\prime}_{-}$
in $P^{\prime}$ to $R_{+}(N^{\prime})$ (to get a sutured manifold homeomorphic
to $N^{\prime}$) followed by attaching the product over $(F_{P})_{-}$ in
$P^{\prime}$ along their sutures.

Now we observe that $C^{\prime}$ is indeed a compression body and we determine
its handle number. Since $P^{\prime}$ is a product with
$R_{-}(P^{\prime})=\Sigma^{\prime}_{-}\cup(F_{P})_{-}$, it splits along the
vertical disks and annuli over the curves
$\partial\Sigma^{\prime}_{-}\cap\partial(F_{P})_{-}$ as a product over
$\Sigma^{\prime}_{-}$ and a product over $(F_{P})_{-}$. The curves
$\partial\Sigma^{\prime}_{-}\cap\partial(F_{P})_{-}$ which are not in the
sutures of $P^{\prime}$ however are identified with
$\partial\Sigma^{\prime}_{+}\cap(F_{N})_{-}$ where they do belong to the
sutures of $N^{\prime}$. Thus we may regard $C^{\prime}$ as being obtained by
attaching the product over $\Sigma^{\prime}_{-}$ in $P^{\prime}$ to
$R_{+}(N^{\prime})$ (to get a sutured manifold homeomorphic to $N^{\prime}$)
followed by attaching the product over $(F_{P})_{-}$ in $P^{\prime}$ along
their sutures. Hence $C^{\prime}$ is sutured manifold homeomorphic to the
sutured manifold $N^{\prime}$ extended along its sutures by a product. See
Figure 12. Thus $h(C^{\prime})=h(N^{\prime})$. Moreover, since $N^{\prime}$ is
a compression body, so is $C^{\prime}$.

As $h(N^{\prime})$ is either $h(N)-2$ or $h(N)$ depending on whether or not
$N$ is a handlebody by Lemma 5.2, we have our result about handle numbers. The
result for handle index immediately follows. ∎

###  5.2. An Upper Bound

######  Theorem 5.4.

Let $(M,\gamma)$ be a connected sutured manifold without toroidal sutures. For
any non-zero class $\xi\in H_{2}(M,\partial M)$, there is a surface $F$
representing $\xi$ so that $(M,\gamma)$ has a Heegaard splitting $(M,F,S)$
such that $h(M,F,S)=h(M,\gamma,0)-2d$. Consequently,

$h(M,\gamma,\xi)\leq h(M,\gamma,0)-2d.$

Equivalently,

$j(M,\gamma,\xi)\leq j(M,\gamma,0).$

Recall that $d=0$, $1$, or $2$ depending on whether neither, just one, or both
of $R_{+}$ and $R_{-}$ are empty and that $h(M,\gamma,0)$ is the linear handle
number of $(M,\gamma)$.

###### Proof.

Since $(M,\gamma)$ has no toroidal sutures, every $\xi\in H_{2}(M,\partial M)$
is regular.

Let $(M,\Sigma)=(A,B;\Sigma)$ be a Heegaard splitting of the sutured manifold
$(M,\gamma)$. So the splitting has handle number $h(M,\Sigma)=h(A)+h(B)$.

By Theorem 3.6, there is a conditioned surface $F_{0}$ representing $\xi$ that
realizes $h(M,\gamma,\xi)$. By a small isotopy, one may assume it is
transverse to $\Sigma$.

In particular, $\partial F_{0}$ intersects $A(\gamma)$ in essential simple
closed curves and coherently oriented spanning arcs. Since $\Sigma$ is a
Heegaard surface, $\partial\Sigma$ is the union of the core curves of
$A(\gamma)$. So for any given component of $\partial\Sigma$, its intersections
with $\partial F_{0}$ are all of the same sign. Hence if a non-empty
collection of components of $F_{0}\cap\Sigma$ is trivial in
$H_{1}(\Sigma,\partial\Sigma)$, the conditioning of $F_{0}$ implies that the
collection must be disjoint from the boundary. Thus we may further assume
$F_{0}$ has been chosen so that no non-empty submanifold of $F_{0}\cap\Sigma$
is trivial in $H_{1}(\Sigma,\partial\Sigma)$. For example, one may modify an
initial conditioned surface by taking its double curve sum with a collection
of parallel copies of $\Sigma$.

For each $C=A,B$, the restriction of $F_{0}$ to $C$ is a conditioned surface
$F_{0}\cap C$. Then Theorem 5.3 implies there is a conditioned surface $F_{C}$
in $C$ homologous to $F_{0}\cap C$ with $\partial(F_{0}\cap
C)\cap\Sigma=\partial F_{C}\cap\Sigma$ so that the sutured manifold
decomposition of $C$ along $F_{C}$ is a compression body $C^{\prime}$ with
$h(C^{\prime})=h(C)$ or $h(C^{\prime})=h(C)-2$ if $C$ is a handlebody. Recall
that a connected compression body is a handlebody exactly when
$\partial_{-}(C)=\emptyset$.

So then let $F=F_{A}\cup F_{B}$. Observe that by construction $F$ is
homologous to $F_{0}$ so that it represents $\xi$, and moreover
$F\cap\Sigma=F_{0}\cap\Sigma$. Let $S$ be the surface
$F\mathbin{\rotatebox[origin={c}]{-30.0}{$\asymp$}}\Sigma$ pushed off $F$.
Explicitly, we may take a closed collar neighborhood
$N_{F}=F\times[-\epsilon,\epsilon]$ of $F$ so that it intersects $\Sigma$ in a
collar of $F\cap\Sigma$. Then
$S=F_{A}\times\\{-\epsilon\\}\cup(\Sigma\backslash N_{F})\cup
F_{B}\times\\{+\epsilon\\}$ is a properly embedded oriented surface disjoint
from $F$. We then claim that $(M,F,S)$ is a circular Heegaard splitting with
handle number $h(A^{\prime})+h(B^{\prime})$. Here $A^{\prime}$ and
$B^{\prime}$ are the compression bodies obtained by decomposing $A$ and $B$
along $F_{A}$ and $F_{B}$.

Since $\partial\Sigma$ is the collection of core curves of the annular sutures
of $M$, we may regard the compression bodies $A^{\prime}$ and $B^{\prime}$ as
being the components on either side of $\Sigma$ in
$(M\backslash\Sigma)\backslash N_{F}$ or equivalently in $(M\backslash
N_{F})\backslash\Sigma$. Then as $F$ splits $N_{F}$ into
$F\times[-\epsilon,0]$ and $F\times[0,\epsilon]$, we observe that $S$ splits
$M\backslash F$ into the two manifolds $\overline{A^{\prime}}=A^{\prime}\cup
F\times[0,\epsilon]$ and $\overline{B^{\prime}}=B^{\prime}\cup
F\times[-\epsilon,0]$. See Figure 13.

Let us examine $\overline{A^{\prime}}$. Partitioning $F\times[0,\epsilon]$
along the vertical disks and annuli of its intersection with $\Sigma$, we have
$\overline{A^{\prime}}=(A^{\prime}\cup
F_{A}\times[0,\epsilon])\cup(F_{B}\times[0,\epsilon])$. Since $A^{\prime}\cup
F_{A}\times[0,\epsilon]$ is equivalent to $A^{\prime}$, we see that
$\overline{A^{\prime}}$ decomposes along these vertical disks and annuli into
$A^{\prime}$ and the product $F_{B}\times[0,\epsilon]$. Hence
$\overline{A^{\prime}}$ is a compression body with
$h(\overline{A^{\prime}})=h(A^{\prime})$. A similar argument shows that
$\overline{B^{\prime}}$ is a compression body with
$h(\overline{B^{\prime}})=h(B^{\prime})$. Thus $(M,F,S)$ is a circular
Heegaard splitting that divides $M$ into the compression bodies
$\overline{A^{\prime}}$ and $\overline{B^{\prime}}$. Hence
$h(M,F,S)=h(\overline{A^{\prime}})+h(\overline{B^{\prime}})=h(A^{\prime})+h(B^{\prime})$.

By Theorem 5.3, $h(A^{\prime})=h(A)$ or $h(A)-2$ depending on whether or not
$R_{-}(\gamma)$ is empty. Similarly $h(B^{\prime})=h(B)$ or $h(B)-2$ depending
on whether or not $R_{+}(\gamma)$ is empty. Therefore
$h(M,F,S)=h(A)+h(B)-2d=h(M,\Sigma)-2d$. So $h(M,\gamma,\xi)\leq
h(M,F,S)=h(M,\Sigma)-2d$.

Consequently if $(M,\Sigma)$ realizes the linear handle number
$h(M,\gamma,0)$, then we have our claimed result that

$h(M,\gamma,\xi)\leq h(M,\gamma,0)-2d.$

In terms of handle index, this becomes

$j(M,\gamma,\xi)\leq j(M,\gamma,0).$

∎

######  Remark 5.5.

Note that in the proof above, even if $\Sigma$ is connected, both
$\overline{A^{\prime}}$ and $\overline{B^{\prime}}$ may be disconnected when
$F$ is separating in both $A$ and $B$. Furthermore $\overline{A^{\prime}}$ and
$\overline{B^{\prime}}$ may have components that are products, i.e. trivial
compression bodies.

Figure 13. In the top row, we start with a Heegaard surface $\Sigma$ splitting
$(M,\gamma)$ into compression bodies $A,B$. The surface $F$ restricted to each
compression body $A,B$ decomposes it into a new compression body
$A^{\prime},B^{\prime}$. In the bottom row, $\Sigma$ is replaced with
$S=F\mathbin{\rotatebox[origin={c}]{-30.0}{$\asymp$}}\Sigma$. Then $S$ is a
Heegaard surface for the sutured manifold $(M^{\prime},\gamma^{\prime})$
obtained by decomposing $(M,\gamma$) along $F$. The surface $S$ splits
$(M^{\prime},\gamma^{\prime})$ into compression bodies
$\overline{A^{\prime}},\overline{B^{\prime}}$.

##  6\. A lower bound for sutured manifolds without toroidal sutures

###  6.1. Sutured Characteristic

A sutured surface is a compact oriented surface $F$ with a disjoint pair of
submanifolds $r_{+}$ and $r_{-}$ of $\partial F$ such that

  * •

the orientation of $r_{+}$ is consistent with the boundary orientation of
$\partial F$ while the orientation of $r_{-}$ is reversed,

  * •

$\partial F\backslash(r_{+}\cup r_{-})$ is a collection of intervals and
circles, collectively called the sutures, and

  * •

each interval suture joins a component of $\partial r_{+}$ to a component of
$\partial r_{-}$.

At times it is convenient to use $\partial_{+}F$ and $\partial_{-}F$ to refer
to the $1$–manifolds $r_{+}$ and $r_{-}$, $\partial_{v}F$ (for vertical
boundary) to refer to the interval sutures, and $\partial_{0}F$ for the circle
sutures.

We now define a modified Euler characteristic for sutured surfaces. When the
sutured surface $F$ is connected, we define the sutured characteristic of $F$
to be

$\chi_{sut}(F)=-\chi(F)+d(F)+\frac{1}{2}\\#\partial_{v}F+\\#\partial_{0}F.$

where $d(F)=0$ if $\partial_{0}F\neq\emptyset$ and otherwise counts how many
of $r_{+}$ and $r_{-}$ are empty. That is, if $\partial_{0}F=\emptyset$ then
$d(F)=0$, $1$, or $2$ depending on whether neither, just one, or both of
$r_{+}$ and $r_{-}$ are empty. When $F$ has multiple components, then its
sutured characteristic is the sum of $\chi_{sut}$ of its components.

A sutured surface $F$ without circle sutures can be viewed as a cobordism from
$\partial_{-}F$ to $\partial_{+}F$, built from attaching $1$–handles to a
collar of $\partial_{-}F$ and a collection of $0$–handles followed by
attaching $2$–handles as needed. Let $h^{i}(F)$ is the minimum number of
$2$–dimensional $i$–handles used in such a handle structure.

######  Lemma 6.1.

For a sutured surface without circle sutures and an associated minimal handle
structure

$\chi_{sut}(F)=-\chi(F)+h^{0}(F)+h^{2}(F)+\frac{1}{2}\\#\partial_{v}F=h^{1}(F).$

###### Proof.

It suffices to prove this lemma for connected $F$.

Since $\partial_{0}F=\emptyset$ by assumption, $d(F)$ counts how many of
$r_{+}$ and $r_{-}$ are empty. If $r_{-}=\emptyset$, then there must be
$0$–handles used in its handle structure. Since $F$ is connected, a minimal
handle structure has just one $0$–handle. Similarly if $r_{+}=\emptyset$, then
a minimal handle structure has just one $2$–handle. If neither of $r_{+}$ or
$r_{-}$ are empty, then only $1$–handles are needed. This gives the first
equality.

Next, observe that any component of $\partial F$ contains an even number of
interval sutures. One could glue these in pairs to make an oriented surface
${\widehat{F}}$ without sutures. Then we may count that

$\chi({\widehat{F}})=\chi(F)-\frac{1}{2}\\#\partial_{v}F=h^{0}(F)-h^{1}(F)+h^{2}(F),$

which implies the second equality. ∎

######  Lemma 6.2.

Let $F$ be a sutured surface. Then $\chi_{-}(F)\leq\chi_{sut}(F)$.

###### Proof.

Recall that (i) $\chi_{-}$ of a connected surface is just the maximum of
$-\chi$ and $0$ and (ii) $\chi_{-}$ is additive on disjoint unions. Therefore
we may restrict attention to the case that $F$ is connected. If $\chi(F)\leq
0$ then the result is immediate since $d(F)$, $\\#\partial_{v}F$, and
$\\#\partial_{0}F$ are all non-negative. If $\chi(F)>0$ so that
$\chi_{-}(F)=0$, then $F$ is either a sphere $S^{2}$ or disk $D^{2}$ since $F$
is orientable. Since $S^{2}$ is closed, it has $d=2$. So
$\chi_{sut}(S^{2})=0$. For the disk however, $\chi_{sut}$ depends on the
suture structure. Let $D^{2}_{n}$ denote the disk with $2n$ interval sutures,
for $n\geq 0$. Then one finds that $\chi_{sut}(D^{2}_{0})=0$ while
$\chi_{sut}(D^{2}_{n})=n-1$ if $n\geq 1$. Lastly, a disk whose boundary is a
circular suture also has $\chi_{sut}=0$. ∎

Any decomposing surface $F$ in a sutured manifold $(M,\gamma)$ inherits the
structure of a sutured surface: $r_{\pm}=\partial F\cap R_{\pm}$ while the
sutures of $F$ are the intersections of $\partial F$ with the sutures of
$(M,\gamma)$. Thus we may define

$\chi_{(M,\gamma)}(F)=\chi_{sut}(F)$

where $F$ has the induced sutured structure. Then, for any class $\xi\in
H_{2}(M,\partial M)$, we may define its sutured characteristic to be

$\chi_{(M,\gamma)}(\xi)=\min_{F\in\xi}\\{\chi_{(M,\gamma)}(F)\\}$

where the minimum is taken among decomposing surfaces $F$ that represent
$\xi$.

######  Remark 6.3.

Since $\chi_{sut}$ is additive on disjoint unions of surfaces, it follows that
for any class $\xi\in H_{2}(M,\partial M)$

$\chi_{(M,\gamma)}(n\xi)=|n|\chi_{(M,\gamma)}(\xi)\mbox{ for all integers
}n>0.$

######  Lemma 6.4.

For a sutured manifold $(M,\gamma)$, the Thurston norm of a class $\xi\in
H_{2}(M,\partial M)$ is a lower bound for its sutured characteristic. That is,
$x(\xi)\leq\chi_{(M,\gamma)}(\xi)$.

###### Proof.

Suppose a properly embedded surface $F$ in $(M,\gamma)$ represents $\xi$ so
that $\chi_{(M,\gamma)}(\xi)=\chi_{(M,\gamma)}(F)$. Then by Lemma 6.2,
$\chi_{(M,\gamma)}(F)\geq\chi_{-}(F)$. Since $\chi_{-}(F)\geq x([F])=x(\xi)$,
the result follows. ∎

###  6.2. A Lower Bound

Recall, we say a surface $F$ representing a regular class $\xi$ realizes
$h(M,\gamma,\xi)$ if there is a surface $S$ in $M$ such that
$h(M,\gamma,\xi)=h(M,F,S)$. Then the handle sutured characteristic of a
regular class $\xi$ is

$\chi_{(M,\gamma)}^{h}(\xi)=\min_{F\in\xi}\\{\chi_{(M,\gamma)}(F)|F\mbox{
realizes }h(M,\gamma,\xi)\\}.$

By Lemma 3.3, $h(M,\gamma,0)$ is realized by a splitting with $F=\emptyset$.
So for the trivial class $\xi=0$, we have $\chi_{(M,\gamma)}^{h}(0)=0$ since
$\chi_{sut}(\emptyset)=0$.

######  Theorem 6.5.

Let $(M,\gamma)$ a sutured manifold without toroidal sutures. Then for any
$\xi\in H_{2}(M,\partial M)$ the following inequality holds:

$h(M,\gamma,0)-2d\leq h(M,\gamma,\xi)+2\chi_{(M,\gamma)}^{h}(\xi)$

where $d=0,1,2$ depending on whether neither, just one, or both of
$R_{+}(\gamma)$ and $R_{-}(\gamma)$ are empty.

###### Proof.

Since $\chi_{(M,\gamma)}^{h}(\xi)\geq 0$ for all $\xi$, the conclusion is
trivially satisfied if $\xi=0$. So assume $\xi\neq 0$.

Given $(M,\gamma,\xi)$, let $F$ be a decomposing surface representing $\xi$
and realizing $h(M,\gamma,\xi)$ with
$\chi_{(M,\gamma)}(F)=\chi_{(M,\gamma)}^{h}(\xi)$. In $M$, let
$F\times[-\epsilon,\epsilon]$ be a collar of $F$, so that
$M^{\prime}=M\backslash F$ with the sutured manifold decomposition
$(M,\gamma)\overset{F}{\leadsto}(M^{\prime},\gamma^{\prime})$. Recall, we
identify $F$ with $F\times\\{0\\}$ and set $F_{\pm}=F\times\\{\mp\epsilon\\}$.
With this arrangement, the normal of the oriented surface $F_{+}$ points out
of $M\backslash F$ and of $F_{-}$ points into $M\backslash F$. Next choose the
surface $S$ so that $(M,F,S)=(A,B;F,S)$ a circular Heegaard splitting with
handle number $h(M,F,S)=h(A)+h(B)$ realizing $h(M,\gamma,F)$, where:

  * •

$\partial_{-}A=R^{\prime}_{-}=F_{-}\cup(R_{-}\backslash F)$

  * •

$\partial_{+}A=\partial_{-}B=S$

  * •

$\partial_{+}B=R^{\prime}_{+}=F_{+}\cup(R_{+}\backslash F)$

For each $\pm=+,-$, define the collars $C_{\pm}=R_{\pm}\times I$ and
$C^{\prime}_{\pm}=R^{\prime}_{\pm}\times I$ so that $\partial R_{+}\times
I\cup\partial R_{-}\times I=A(\gamma)$ and $\partial R^{\prime}_{+}\times
I\cup\partial R^{\prime}_{-}\times I=A(\gamma^{\prime})$.

Let us first assume that no component of $\partial F$ is contained in the
sutures. Thereafter we will address the necessary modifications when some
component of $\partial$ is contained in the sutures.

Case 1: No component of $\partial F$ is contained in the sutures.
Let us also fix a minimal handle structure on $F$, viewed as a cobordism from
$F\cap R_{-}$ to $F\cap R_{+}$ due to its embedding in the sutured manifold
$(M,\gamma)$. In particular, a component of $F$ has a $0$–handle only if it is
disjoint from $R_{-}$, and it has a $2$–handle only if it is disjoint from
$R_{+}$. Furthermore $h^{1}(F)=\chi_{sut}(F)=\chi_{(M,\gamma)}(F)$.

First observe that $A$ has a compression body structure consisting of
$1$–handles attached to $C^{\prime}_{-}=R^{\prime}_{-}\times I$. (No component
of $A$ is a handlebody since the circular splitting realizes $h(M,\gamma,\xi)$
for $\xi\neq 0$; see Section 2.6.) Since $R^{\prime}_{-}=(R_{-}\backslash
F)\cup F_{-}$, let us first thicken $F_{-}$ by attaching $F\times[0,\epsilon]$
to $A$ and then viewing the collar $C^{\prime}_{-}$ as being $(R_{-}\backslash
F)\times I\cup F\times[0,\epsilon]$. Now we may drag the feet of the
$1$–handles of $A$ so that each is attached to the portion of the collar over
$R_{-}\backslash F$ or a thickened disk neighborhood of a $0$–handle of
$F_{-}$ in $F\times[0,\epsilon]$. Recall that if a component of $F$ has no
$0$–handles in its handle structure it meets $R_{-}$; hence the foot of a
$1$–handle of $A$ over such a component could be dragged to any component of
$R_{-}\backslash F$ that it meets.

Next, consider the manifold $\bar{A}=(A\cup F\times[0,\epsilon])\backslash
C_{+}$ obtained by scalloping out whatever intersects the collar $C_{+}$ of
$R_{+}$. Such intersections are just thickenings of
$(\partial_{+}F)\times[0,\epsilon]$ and their removal does not affect the
aforementioned handle structure of $A$ or the handle structure of $F$. Now let
us consider the handle structure of $\bar{A}$ induced from the $1$–handles of
$A$ and the handles of $F\times[0,\epsilon]\backslash C_{+}$ built upon only
$(R_{-}\backslash F)\times I$ and the thickened $0$–handles of $F$,
${\mathcal{H}}^{0}(F)\times[0,\epsilon]$. Indeed, any $2$–dimensional
$k$–handle of $F$ thickens to a $3$–dimensional $k$–handle of
$F\times[0,\epsilon]$ and also of $F\times[0,\epsilon]\backslash C_{+}$. If a
component of $F$ has no $0$–handles, then its thickened $1$–handles will be
attached to the components of $(R_{-}\backslash F)\times I$ that it meets;
otherwise its thickened $1$–handles will be attached to its thickened
$0$–handles. Observe that the $2$–handles of $\bar{A}$ are the thickened
$2$–handles of $F$, specifically ${\mathcal{H}}^{2}(F)\times[0,\epsilon]$.

At this point, it is helpful to acknowledge that the analogous dual
construction may be carried out for the compression body $B$. That is, we
obtain a handle structure on the manifold $\bar{B}=(B\cup
F\times[-\epsilon,0])\backslash C_{-}$ built upon $(R_{+}\backslash F)\times
I$ and the thickened $2$–handles of $F$,
${\mathcal{H}}^{2}(F)\times[-\epsilon,0]$. Cutting out $C_{-}$ just removes
thickenings of $(\partial_{-}F)\times[-\epsilon,0]$. And any $2$–dimensional
$k$–handle of $F$ thickens to a $3$–dimensional $k+1$–handle of
$F\times[-\epsilon,0]$ which is a dual $2-k$–handle. Indeed, the $1$–handles
of $\bar{B}$ (the dual $2$–handles) are the thickened $0$–handles of $F$,
specifically ${\mathcal{H}}^{0}(F)\times[-\epsilon,0]$.

So now we may consider that $\bar{A}\cup C_{-}$ only adds to $\bar{A}$ the
thickenings of $(\partial_{-}F)\times[-\epsilon,0]$ to complete
$(R_{-}\backslash F)\times I$ to $C_{-}=R_{-}\times I$. Hence $\bar{A}\cup
C_{-}$ has the handle structure of $\bar{A}$ built upon $C_{-}$ and the
$0$–handles ${\mathcal{H}}^{0}(F)\times[0,\epsilon]$. By deleting the
$2$–handles ${\mathcal{H}}^{2}(F)\times[0,\epsilon]$, this becomes a
compression body, though one that is possibly disconnected. Similarly
$\bar{B}\cup C_{+}$ is made into a possibly disconnected compression body
built upon $C_{+}$ and the dual $0$–handles
${\mathcal{H}}^{2}(F)\times[-\epsilon,0]$ by deleting its dual $2$–handles,
${\mathcal{H}}^{0}(F)\times[-\epsilon,0]$. Though as these dual $2$–handles
are just $1$–handles, we may add them to the other compression body.
Ultimately, we form the compression bodies

$A^{*}=\bar{A}\cup
C_{-}\cup({\mathcal{H}}^{0}(F)\times[-\epsilon,0])\backslash({\mathcal{H}}^{2}(F)\times[0,\epsilon])$

and

$B^{*}=\bar{B}\cup
C_{+}\cup({\mathcal{H}}^{2}(F)\times[0,\epsilon])\backslash({\mathcal{H}}^{0}(F)\times[-\epsilon,0]).$

Since $A^{*}$ and $B^{*}$ are compression bodies with disjoint interiors where
$M=A^{*}\cup B^{*}$, $\partial_{-}A^{*}=R_{-}$, and $\partial_{+}B^{*}=R_{+}$,
they define a Heegaard splitting (and Heegaard surface
$S^{*}=\partial_{+}A^{*}=\partial_{-}B^{*}$) for the connected sutured
manifold $M$ and are hence connected as well. See Figure 14 for a schematic of
this construction.

Figure 14. Schematic pictures for the proof of Theorem 6.5. (a) The sutured
manifold $M$ with the surface $F$ and a collar is indicated. (b) The sutured
manifold $M^{\prime}=M\backslash F$ is obtained by decomposing $M$ along $F$.
(c) The Heegaard surface $S$ for $M^{\prime}$, the compression bodies $A$ and
$B$, and the cores of their $1$–handles and dual $1$–handles are shown. (d)
Halves of the collar of $F$ are added back to $A$ and $B$ to reform $M$, the
collars $C_{+}$ of $R_{+}$ and $C_{-}$ of $R_{-}$ are removed from $A$ and $B$
respectively and then added to the other to form the manifolds
$\overline{A}\cup C_{-}$ and $\overline{B}\cup C_{+}$. In the halves of the
collar of $F$, thickened handle decompositions of $F$ are now indicated. (e)
Reassigning $2$–handles of $\overline{A}\cup C_{-}$ and $1$–handles of
$\overline{B}\cup C_{+}$ associated to handle structures of $F$ induce
compression bodies $A^{*}$ and $B^{*}$. (f) The feet of the $1$–handles of
$A^{*}$ and dual $1$–handles of $B^{*}$ are pulled to lie in $R_{-}$ and
$R_{+}$ respectively.

As built, $A^{*}$ has $h^{0}(F)$ $0$–handles and $h^{1}(A)+h^{1}(F)+h^{0}(F)$
$1$–handles. These $0$–handles may each be cancelled with a $1$–handle, unless
$R_{-}=\emptyset$ in which case $A^{*}$ must be a handlebody and one
$0$–handle must remain. This induces $A^{*}$ with a handle structure
consisting of $h^{0}(A^{*})$ $0$–handles and $h^{1}(A)+h^{1}(F)+h^{0}(A^{*})$
$1$–handles. Similarly all of the dual $0$–handles of $B^{*}$ may be cancelled
with dual $1$–handles, unless $R_{+}=\emptyset$ in which case $B^{*}$ is a
handlebody and one dual $0$–handle must remain. Therefore we obtain a Heegaard
splitting $(M,A^{*},B^{*})$ with handle number

$\displaystyle h(M,A^{*},B^{*})$ $\displaystyle=h(A^{*})+h(B^{*})$
$\displaystyle=(h^{0}(A^{*})+h^{1}(A^{*}))+(h^{0}(B^{*})+h^{1}(B^{*}))$
$\displaystyle=(h^{0}(A^{*})+h^{1}(A)+h^{1}(F)+h^{0}(A^{*}))+(h^{0}(B^{*})+h^{1}(B)+h^{1}(F)+h^{0}(B^{*}))$
$\displaystyle=h^{1}(A)+h^{1}(B)+2h^{1}(F)+2(h^{0}(A^{*})+h^{0}(B^{*}))$
$\displaystyle=h(M,\gamma,\xi)+2\chi_{sut}(F)+2d.$

Therefore

$h(M,\gamma,0)\leq h(M,\gamma,\xi)+2\chi_{(M,\gamma)}(F)+2d.$

Thus, since $\chi_{(M,\gamma)}(F)=\chi_{(M,\gamma)}^{h}(\xi)$, we have the
desired inequality.

Case 2: Some component of $\partial F$ is contained in the sutures.
We will perturb the surface $F$ to a surface $\tilde{F}$ such that

  * •

$\tilde{F}$ is a decomposing surface representing $\xi$ and realizing
$h(M,\gamma,\xi)$,

  * •

$\chi_{(M,\gamma)}(\tilde{F})=\chi_{(M,\gamma)}(F)$, and

  * •

no component of $\partial\tilde{F}$ is contained in the sutures.

Then we may apply the argument of Case 1 to $\tilde{F}$ to get the desired
result.

For each component of $A(\gamma)$ that contains a component of $\partial F$,
isotop $F\cup S$ in a collar of the annular suture (taken in $(M,\gamma)$) so
that afterwards each component of $\partial F$ and $\partial S$ in this collar
now meets the annular suture in a pair of spanning arcs and each $R_{+}$ and
$R_{-}$ in a single arc. See the top row of Figure 15. Let $\tilde{F}$ and
$\tilde{S}$ be the surfaces that result from these isotopies. Since
$\tilde{F}$ is isotopic to $F$, it too represents $\xi$.

Observe that since each circular suture of $F$ has been perturbed into a pair
of vertical sutures, the perturbed surface $\tilde{F}$ now has no circular
sutures while $\chi_{(M,\gamma)}(\tilde{F})=\chi_{(M,\gamma)}(F)$.

Figure 15. The top row starts with the isotopy of the surfaces $F$ and $S$ in
a neighborhood of an annular suture that contains components of $\partial F$
(and hence also $\partial S$) to the perturbed surfaces $\tilde{F}$ and
$\tilde{S}$. Thereafter an ambient isotopy straightens out $\tilde{F}$ and
$\tilde{S}$ at the expense of perturbing the annular suture. The bottom row
shows that the decomposition along $F$ and how it contains $S$ is isotopic to
the decomposition along $\tilde{F}$ and how it contains $\tilde{S}$.

Moreover, as the perturbation of $(M,F,S)$ to $(M,\tilde{F},\tilde{S})$ occurs
within a collar of $A(\gamma)$, these splittings are basically equivalent:
Shaving off a collar of $A(\gamma)$ from each $(M,F,S)$ and
$(M,\tilde{F},\tilde{S})$ leaves equivalent manifolds. Ultimately, we just
need to compare the decompositions along each $F$ and $\tilde{F}$ in this
collar. Figure 15 shows that these decompositions leave isotopic sutured
manifolds $(M\backslash F,\gamma^{\prime})$ and
$(M\backslash\tilde{F},\tilde{\gamma})$ with surfaces $S$ and $\tilde{S}$ that
are equated by such an isotopy. ∎

######  Theorem 6.6.

Let $(M,\gamma)$ a sutured manifold without toroidal sutures. Then for any
$\xi\in H_{2}(M,\partial M)$ we have

$h(M,\gamma,\xi)\leq h(M,\gamma,0)-2d\leq
h(M,\gamma,\xi)+2\chi^{h}_{(M,\gamma)}(\xi)$

where $d=0,1,2$ depending on whether neither, just one, or both of
$R_{+}(\gamma)$ and $R_{-}(\gamma)$ are empty.

##  7\. Bounds for knots and links

###  7.1. Tunnel numbers and Morse Novikov numbers of links

Let $M$ be a connected, compact oriented $3$–manifold with non-empty boundary.
A tunnel system for $M$ is a properly embedded collection of arcs $\tau$ so
that $M-\mathcal{N}(\tau)$ is a handlebody. The tunnel number ${\rm{TN}}(M)$
of $M$ is the minimum cardinality of its tunnel systems. Equivalently, if we
regard $M$ as a connected sutured manifold $(M,\gamma_{+})$ for which
$\partial M=R_{+}$, then the tunnel number of $M$ is the minimum of $h(B)$
among Heegaard splittings $(A,B;S)$ for $M$. This gives the equality

${\rm{TN}}(M)+g(M,\gamma_{+})+1=h(M,\gamma_{+},0).$

######  Observation 7.1.

For manifolds $M$ with non-empty toroidal boundary,
$h(M,\gamma_{+},0)=2{\rm{TN}}(M)+2$.

Let $L$ be a link in a closed $3$–manifold $Y$. The exterior of $L$ is the
complement $M=Y\backslash L$ of a regular neighborhood. Let $(M,\gamma_{+})$
be obtained by setting $R_{+}(\gamma_{+})=\partial M$. The tunnel number
${\rm{TN}}(L)$ of $L$ is ${\rm{TN}}(M,\gamma_{+})$.

Further suppose $L$ is an oriented, null-homologous link in $Y$. Let
$(M,\gamma_{0})$ be obtained by setting $T(\gamma_{0})=\partial M$. Then its
Seifert surfaces define a regular class $\xi\in H_{2}(M,\partial M)$. The
Morse-Novikov number ${\rm{MN}}(L)$ of $L$ is
${\rm{MN}}(L)=h(M,\gamma_{0},\xi)$. (Note that this definition can be extended
to rationally null-homologous links as well.) By Theorem 4.1,
$h(M,\gamma_{0},\xi)=h(M,\gamma_{+},\xi)$. Hence we also have
${\rm{MN}}(L)=h(M,\gamma_{+},\xi)$.

###  7.2. Bounds for knots and links in $S^{3}$.

Let $L$ be an oriented link in $S^{3}$ of $|L|$ components. A Seifert surface
for $L$ is an oriented surface $F$ possibly disconnected but without closed
components such that $\partial F=L$. Let $M=S^{3}\backslash L$ be the exterior
of $L$, and consider the sutured structure $(M,\gamma_{+})$ obtained by
setting $R_{+}(\gamma_{+})=\partial M$.

######  Corollary 7.2 ([Paj10, Theorem 1.1]).

The Morse-Novikov number of an oriented link $L$ in $S^{3}$ is at most twice
its tunnel number. That is

${\rm{MN}}(L)\leq 2{\rm{TN}}(L).$

###### Proof.

By Observation 7.1, $h(M,\gamma_{+},0)=2{\rm{TN}}(L)+2$. Now for the homology
class $\xi$ of a Seifert surface for $L$ restricted to $(M,\gamma_{+})$,
Theorem 5.4 implies that $h(M,\gamma_{+},\xi)\leq h(M,\gamma_{+},0)-2$. Since
${\rm{MN}}(L)=h(M,\gamma_{+},\xi)$, this gives our result. ∎

######  Lemma 7.3.

Suppose a surface $F$ realizes $h(M,\gamma_{+},\xi)$ and
$\chi_{(M,\gamma_{+})}^{h}(\xi)$. Then $F$ is a (possibly disconnected)
Seifert surface for $L$ and

$\chi_{(M,\gamma_{+})}(\xi)=2g(F)+|L|-|F|.$

###### Proof.

Since $M$ is the exterior of a link in $S^{3}$, any closed component of $F$
homologically trivial in $H_{2}(M,\partial M)$. Then by Lemma 3.3,
$h(M,\gamma_{+},\xi)$ is realized by a subsurface of $F$ without closed
components. Since $F$ already realizes $\chi_{(M,\gamma_{+})}^{h}(\xi)$, it
must have no closed components. By applications of Lemma 2.8 along annuli, we
may ensure $\partial F$ has no pairs of parallel but oppositely oriented
components in any component of $\partial M$ while maintaining
$\chi_{(M,\gamma_{+})}^{h}(\xi)$. Then since $\xi$ is a Seifert class,
$\partial F$ must have a single component in each component of $\partial M$.
Hence $F$ is a Seifert surface for $L$.

As a properly embedded surface in $(M,\gamma_{+})$, $F$ inherits the structure
of a sutured surface with $\partial F=\partial_{+}F=r_{+}$ so that
$h^{0}(F)=|F|$, $h^{2}(F)=0$, and $\\#\partial_{v}(F)=0$. Hence by Lemma 6.1,

$\chi_{(M,\gamma_{+})}(F)=h^{1}(F)=|F|-\chi(F).$

If $F$ has components $F_{1},\dots,F_{|F|}$, then

$\chi(F)=\sum\chi(F_{i})=\sum\left(2-2g(F_{i})-|\partial
F_{i}|\right)=2|F|-|\partial F|-2\sum g(F_{i}).$

So we have

$\chi_{(M,\gamma_{+})}(F)=|F|-\chi(F)=|F|-\left(2|F|-|\partial F|-2\sum
g(F_{i})\right)=2g(F)+|L|-|F|$

since $|\partial F|=|L|$. ∎

Define $g_{\rm{MN}}(L)=g(F)$ for a surface $F$ that realizes
$h(M,\gamma_{+},\xi)$ and $\chi_{(M,\gamma_{+})}^{h}(\xi)$. If $F$ has no
closed components, then $F$ is a Seifert surface for $L$.

######  Corollary 7.4.

Let $L$ be an oriented link in $S^{3}$. Then

$2{\rm{TN}}(L)\leq{\rm{MN}}(L)+4g_{{\rm{MN}}}(L).$

###### Proof.

Let $\xi$ be the class of a Seifert surface for $L$ realizing
$h(M,\gamma_{+},\xi)$ and $\chi_{(M,\gamma_{+})}^{h}(\xi)$. By Theorem 6.5,

$h(M,\gamma_{+},0)-2\leq h(M,\gamma_{+},\xi)+2\chi_{(M,\gamma_{+})}^{h}(\xi)$

because $d=1$ as $R_{+}=\partial M$. By definition,
$h(M,\gamma_{+},\xi)={\rm{MN}}(L)$.

By Observation 7.1 $h(M,\gamma_{+},0)=2{\rm{TN}}(L)+2$. Lemma 7.3 and the
definition of $g_{\rm{MN}}(L)$ imply that
$\chi_{(M,\gamma_{+})}^{h}(\xi)=2g_{\rm{MN}}(L)$. Hence
$2{\rm{TN}}(L)\leq{\rm{MN}}(L)+4g_{{\rm{MN}}}(L)$. ∎

######  Lemma 7.5.

If $L$ is a link in $S^{3}$, then ${\rm{MN}}(L)$ is realized by a connected
Seifert surface.

###### Proof.

Suppose a surface $F$ realizes ${\rm{MN}}(L)=h(M,\gamma_{+},\xi)$ where $\xi$
is the Seifert class for $L$. By Lemma 3.3, we may assume no component of $F$
is closed. Thus $F$ is non-separating. So now if $F$ is not connected, then by
Lemma 3.7 there is a sequence of tubings of $F$ that produces a connected
surface that also realizes ${\rm{MN}}(L)$. ∎

######  Theorem 7.6.

For an oriented link $L$ in $S^{3}$,

${\rm{MN}}(L)\leq 2{\rm{TN}}(L)\leq{\rm{MN}}(L)+4g_{{\rm{MN}}}(L).$

###### Proof.

This follows directly from Corollaries 7.2 and 7.4. ∎

######  Corollary 7.7.

If $L$ is a fibered link then ${\rm{TN}}(L)\leq 2g(L)$.

###### Proof.

If $L$ is fibered, then ${\rm{MN}}(L)=0$ and $g_{\rm{MN}}(L)=g(L)$. Then
Theorem 7.6 implies the result. ∎

######  Corollary 7.8.

Suppose $\\{K_{k}\\}_{k\in\mathbb{N}}$ is a family of knots on which
${\rm{TN}}$ is unbounded. Then at most one of $g_{{\rm{MN}}}$ and ${\rm{MN}}$
is bounded.

###### Proof.

By Theorem 7.6, ${\rm{MN}}(K_{k})+4g_{{\rm{MN}}}(K_{k})\geq
2{\rm{TN}}(K_{k})$. ∎

######  Corollary 7.9.

There is a family of genus one knots $\\{K_{k}\\}_{k\in\mathbb{N}}$ for which
either $g_{{\rm{MN}}}$ is unbounded or ${\rm{MN}}$ is unbounded.

###### Proof.

By [EMnL99, Theorem 3.7(2)], the knots $K_{n}$ described in [EMnL99, Figure 3]
have ${\rm{TN}}(K_{n})\geq n$. One readily observes that these knots are all
genus one knots. Now apply Corollary 7.8. ∎

######  Remark 7.10.

We expect that the knots $K_{n}$ described in [EMnL99, Figure 3] have a unique
incompressible Seifert surface, except possibly for small $n$. If so, then
$g_{MN}(K_{n})=g(K_{n})=1$ for those knots. Corollaries 7.8 and 7.9 would then
imply that $\\{{\rm{MN}}(K_{n})\\}$ is unbounded.

######  Remark 7.11.

Hirasawa-Rudolph asked for examples of knots where ${\rm{MN}}(K)>2g(K)$
[HR03]. The only explicit examples known so far are certain pretzel knots with
$g=1$ and ${\rm{MN}}=4$ [MGNnRL15]. These knots also have $g_{\rm{MN}}=1$.

######  Conjecture 7.12 (Cf. [MG09, Remark p440]).

For an oriented link $L$ in $S^{3}$, $g_{{\rm{MN}}}(L)=g(L)$.

##  8\. The circular handle number function on homology

###  8.1. Regular classes

Recall that for a sutured manifold $(M,\gamma)$, a class $\xi\in
H_{2}(M,\partial M)$ is regular if every surface representing $\xi$ meets
every toroidal suture. A class is non-regular if it is not regular. To phrase
this more homologically, let $T(\gamma)\subset\partial M$ be the toroidal
sutures of $(M,\gamma)$. Then a class $\xi$ is non-regular if $\partial\xi\in
H_{1}(\partial M)$ is zero on some component of $T(\gamma)$. This second
version of the definition extends to other coefficients. In particular, we may
now speak of regular classes and non-regular classes of $H_{2}(M,\partial
M;\mathbb{R})$.

######  Lemma 8.1.

Let $(M,\gamma)$ be a connected sutured manifold. Unless $M$ is a rational
homology sphere, the set of non-regular classes of $H_{2}(M,\partial
M;\mathbb{R})$ is a finite union of subspaces of positive codimension.

###### Proof.

Since $H_{2}(M,\partial M;\mathbb{R})=0$ exactly when $M$ is a rational
homology sphere, let us assume $H_{2}(M,\partial M;\mathbb{R})\neq 0$. If
$T(\gamma)=\emptyset$, then every non-zero class is regular and the result
holds. So now further assume $T(\gamma)\neq\emptyset$.

Let $T$ be a non-empty subset of the components of $T(\gamma)$. Then we claim
that the set $V_{T}$ of classes of $H_{2}(M,\partial M;\mathbb{R})$ whose
boundary vanishes on $T$ form a subspace. By definition, the members of
$V_{T}$ are necessarily non-regular classes. So now suppose $\sigma$ and
$\tau$ are non-regular classes where $\partial\sigma$ and $\partial\tau$ are
zero on $T$. Since $\partial(r\sigma+s\tau)=r\partial\sigma+s\partial\tau$ for
any $r,s\in\mathbb{R}$, $\partial(r\sigma+s\tau)$ is zero on $T$. Hence
$r\sigma+s\tau\in V_{T}$. Thus $V_{T}$ is a subspace.

Consequently, letting $\mathcal{T}$ be the collection of all non-empty subsets
of components of $T(\gamma)$, the set of non-regular classes of
$H_{2}(M,\partial M;\mathbb{R})$ is $\bigcup_{T\in\mathcal{T}}V_{T}$. Since
$T(\gamma)$ is finite, $\mathcal{T}$ is finite, so this is a finite union.

Finally, we claim that there exists a regular class. By the Half-Lives Half-
Dies Lemma (that half of the rank of $H_{1}(\partial M;\mathbb{Z})$ dies upon
the inclusion of $\partial M$ into $M$), for each component of $\partial M$
there is a primitive class $\eta\in H_{2}(M,\partial M;\mathbb{Z})$ for which
$\partial\eta$ is non-zero on that component. In particular, for each torus
$T_{i}$ of $T(\gamma)$ let $\eta_{i}$ be such a primitive class (where $i$
runs from $1$ to $n=|T(\gamma)|$). Then for each $k$ from $1$ to $n$ we may
successively find a non-zero integral coefficient $c_{k}$ so that the class
$\zeta_{k}=\sum_{i=1}^{k}c_{i}\eta_{i}$ has $\partial\zeta_{k}$ non-zero on
each of $T_{1},\dots,T_{k}$. Thus $\zeta_{n}$ will be a regular class. ∎

######  Lemma 8.2.

For a connected sutured manifold $(M,\gamma)$, the set of regular classes of
$H_{2}(M,\partial M;\mathbb{R})$ is a dense open set.

###### Proof.

This follows from the non-regular classes being a finite union of subspaces of
positive codimension, Lemma 8.1. ∎

###  8.2. A function on homology

Fix a sutured manifold structure $(M,\gamma)$ on a $3$–manifold $M$. To begin,
let us write $h(\xi)=h(M,\gamma,\xi)$ for an integral regular class $\xi\in
H_{2}(M,\partial M;\mathbb{Z})$. Theorem 8.3 examines properties of this
function. We extend $h$ to real and non-regular classes in Theorem 8.4.

######  Theorem 8.3.

For a sutured manifold $(M,\gamma)$, the handle number function

$h\colon H_{2}(M,\partial M)\to\mathbb{N}$

has the following properties for any regular classes $\alpha,\beta\in
H_{2}(M,\partial M)$:

  1. (1)

$h(\beta)\leq N$ for some constant $N$ (depending on $(M,\gamma)$),

  2. (2)

$h(k\beta)=h(\beta)$ for any integer $k>0$, and

  3. (3)

$h(\alpha+m\beta)\leq h(\beta)$ for sufficiently large $m$.

###### Proof of Theorem 8.3.

(1) This follows immediately from Theorem 5.4 (and Theorem 4.1 in case
$(M,\gamma)$ has toroidal sutures).

(2) Let $k$ be a positive integer. If $h(k\beta)=h(\beta)$ holds for primitive
homology classes, then it holds for non-primitive ones too. So assume $\beta$
is a primitive homology class.

Say $F$ realizes $h(\beta)$. Then let $F_{k}$ be $k$ parallel copies of $F$
(in a collar neighborhood of $F$ disjoint from $S$), and let $S_{k}$ be $S$
along with $k-1$ copies of $F$ interlaced between the previous stack. Then the
Heegaard splitting $(M,F_{k},S_{k})$ has the same handle number as $(M,F,S)$.
Since $[F_{k}]=k[F]=k\beta$, $h(\beta)\geq h(k\beta)$.

Now for the other direction. Among surfaces realizing $h(k\beta)$, say
$\tilde{F}$ is one with the least number of components. By Theorem 3.6 and its
proof, we may assume $\tilde{F}$ is well-conditioned. In particular,
$[\tilde{F}]=k\beta$, there is a Heegaard splitting $(M,\tilde{F},\tilde{S})$
with $h(M,\tilde{F},\tilde{S})=h(M,\tilde{F})=h(k\beta)$, $\tilde{F}$
satisfies condition (C1), and no other such surface has fewer components. For
each connected compression body $A_{i}$ of the compression body $\tilde{A}$ of
the Heegaard splitting
$(M,\tilde{F},\tilde{S})=(\tilde{A},\tilde{B};\tilde{F},\tilde{S})$, let
$\tilde{F}_{i}=\tilde{F}\cap A_{i}$. Observe that each $\tilde{F}_{i}$ also
satisfies condition (C1) since it is a union of components of $\tilde{F}$.

If $\tilde{F}_{i}$ has just one component for each $i$, then the components of
$M\backslash\tilde{F}$ provide the homology between pairs of components of
$\tilde{F}$. So they all must represent the same integral homology class, say
$\beta^{\prime}$. Also, let $k^{\prime}=|\tilde{F}|$ be the number of these
components. So then we have
$k\beta=[\tilde{F}]=[\cup_{i}\tilde{F}_{i}]=\sum_{i}[\tilde{F}_{i}]=k^{\prime}\beta^{\prime}$.
Since $\beta$ is a primitive class, the Representation Theorem of [MP77]
implies that $\tilde{F}$ has at least $k$ components, ie. $k^{\prime}\geq k$.
Hence $\beta=\frac{k^{\prime}}{k}\beta^{\prime}$ in $H_{2}(M,\partial
M;\mathbb{Q})$. Since $\beta$ is a primitive class and $\beta^{\prime}$ is an
integral class, it follows that in fact $k^{\prime}=k$ and
$\beta^{\prime}=\beta$. That is, each $\tilde{F}_{i}$ represents the primitive
class $\beta$. In this case, let $\tilde{F}_{1}$ be one of these components.
Then by amalgamations along all other components (Lemma 3.1), we obtain a
circular Heegaard splitting $(M,\tilde{F}_{1},\tilde{S}_{1})$ (where
$\tilde{S}_{1}$ is the surface obtained from the amalgamations) that
represents $h(\beta)$.

On the other hand, suppose that, for say $i=1$, we have $|\tilde{F}_{1}|\geq
2$. If $A_{1}$ had a minimal handle structure with a $1$-handle whose feet
were in two components of $\tilde{F}_{1}$, then by Lemma 3.7 a tubing along
the handle would create a Heegaard splitting $(M,\tilde{F}^{*},\tilde{S}^{*})$
realizing $h(k\beta)$ but with $|\tilde{F}^{*}|<|\tilde{F}|$ contrary to
assumptions. Hence the components of $\tilde{F}_{1}$ must all lie in the same
component of $\partial_{-}A_{1}$, and $A_{1}$ is formed from
$\partial_{-}A_{1}\times I$ with just enough $1$–handles to connect these
components of $\partial_{-}A_{1}$. (So $h(A_{1})=|\partial_{-}A_{1}|-1$.)

Say $Q_{0}$ is the component of $\partial_{-}A_{1}$ containing
$\tilde{F}_{1}$. Then $Q_{0}\backslash\tilde{F}_{1}$ must be a collection of
components of $R_{-}\backslash\tilde{F}$ which meet $\tilde{F}_{-}$. (If a
component of $R_{-}\backslash\tilde{F}$ does not meet $\tilde{F}_{-}$, then it
does not belong to a component of $\partial_{-}A$ that meets $\tilde{F}$.)
Since $\tilde{F}_{1}$ has at least $2$ components, $\tilde{F}_{1}\subset
Q_{0}$, and $Q_{0}$ is connected, some component $Q$ of
$Q_{0}\backslash\tilde{F}_{1}$ must meet least two components of
$\tilde{F}_{1}$. Let $R_{0}$ be the component of $R_{-}$ containing this
component $Q$. Then Lemma 3.5 produces a new decomposing surface
$\tilde{F}_{R_{0}}^{\prime}$ homologous to $\tilde{F}$ and maintaining that
$h(M,\tilde{F}_{R_{0}}^{\prime})=h(M,\tilde{F})=h(k\beta)$, but with fewer
components. However, this is contrary to our assumptions.

(3) We will construct a splitting $(M,F,S)$ with $h(M,F,S)\leq h(\beta)$ such
that $F$ represents $\alpha+m\beta$ for some $m>0$.

Suppose $h(\beta)$ is realized by the Heegaard splitting
$(M,F_{\beta},S_{\beta})$ which splits $M$ into the compression bodies
$A_{\beta}$ and $B_{\beta}$. By Lemma 3.3 we may assume that $F_{\beta}$ is
well-conditioned. Hence no component of either $A_{\beta}$ or $B_{\beta}$ is a
handlebody. (If a component were a handlebody, then its boundary would be a
component of $S_{\beta}$ which is a Heegaard surface for a component of
$M\backslash F_{\beta}$. The collection of components of $F_{\beta}$ meeting
that component of $M\backslash F_{\beta}$ would be null homologous.) Suppose
the surface $F_{\alpha}$ represents $\alpha$. Isotop $F_{\alpha}$ to both
minimally intersect and be transverse to $F_{\beta}\cup S_{\beta}$ and
$\gamma$.

Let $(M_{i},\gamma_{i})$ for $i=1,\dots,n$ be the components of the result of
the sutured manifold decomposition of $(M,\gamma)$ along $F_{\beta}$. Then for
each $i$, the surface $S_{i}=S_{\beta}\cap M_{i}$ is a Heegaard surface for
$(M_{i},\gamma_{i})$. Let $F_{i}=F_{\alpha}\cap M_{i}$.

The proof of Theorem 5.4 produces a new surface $F_{i}^{\prime}$ in
$\mathcal{N}(F_{i}\cup S_{i})$ that is homologous there to
$F_{i}\mathbin{\rotatebox[origin={c}]{-30.0}{$\asymp$}}m_{i}S_{i}$ for
sufficiently large $m_{i}>0$, satisfies $F_{i}^{\prime}\cap
R_{\pm}(\gamma_{i})=F_{i}\cap R_{\pm}(\gamma_{i})$, and decomposes the
compression bodies $A_{i}$ and $B_{i}$ of $M_{i}\backslash S_{i}$ into
compression bodies $A_{i}^{\prime}$ and $B_{i}^{\prime}$ of the same handle
numbers.

Choose $m$ to be larger than all of the $m_{i}$, and then update each
$F_{i}^{\prime}$ be the surface as above but using $m-1$ instead of $m_{i}$.
Update $A_{i}^{\prime}$ and $B_{i}^{\prime}$ as well. Then we may set
$F^{\prime}=\cup F_{i}^{\prime}$ so that
$[F^{\prime}]=[F]+(m-1)[S_{\beta}]=\alpha+(m-1)\beta$. Next form the surfaces
$F_{0}=F^{\prime}\mathbin{\rotatebox[origin={c}]{-30.0}{$\asymp$}}F_{\beta}$
and
$S_{0}=F^{\prime}\mathbin{\rotatebox[origin={c}]{-30.0}{$\asymp$}}S_{\beta}$,
and note that $[F_{0}]=[S_{0}]=\alpha+m\beta$. (Indeed,
$[F_{0}]=[F^{\prime}]+[F_{\beta}]=\alpha+(m-1)\beta+\beta=\alpha+m\beta$.)
Observe now that $(M,F_{0},S_{0})$ is a circular Heegaard splitting with
compression bodies $A_{0}=\bigcup A_{i}\cup(F^{\prime}\cap B_{i})\times I$ and
$B_{0}=\bigcup B_{i}\cup(F^{\prime}\cap A_{i})\times I$. Since $h(A_{0})=\sum
h(A_{i})=h(A)$ and $h(B_{0})=\sum h(B_{i})=h(B)$, we find that
$h(M,F_{0},S_{0})=h(M,F_{\beta},S_{\beta})$ which is $h(\beta)$. Hence
$h(\alpha+m\beta)\leq h(\beta)$. ∎

######  Theorem 8.4.

The handle number extends to a function $h\colon H_{2}(M,\partial
M;\mathbb{R})\to\mathbb{N}$ that is

  1. (1)

bounded by constant,

  2. (2)

constant on rays from the origin, and

  3. (3)

locally maximal.

###### Proof.

Let $\beta$ be a non-zero regular class. Since $h(k\beta)=h(\beta)$ for any
positive integer $k$, we may define $h(r\beta)=h(\beta)$ for any positive real
number $r$. Hence $h$ is constant on rays from the origin in $H_{2}(M,\partial
M;\mathbb{R})-\\{0\\}$ that meet integral regular classes.

Let $S$ be the quotient space of rays from the origin in $H_{2}(M,\partial
M;\mathbb{R})-\\{0\\}$. Say a point in $S$ is rational if its ray contains an
integral class; otherwise say it is irrational. Similarly, say a point in $S$
is regular if its ray belongs to the regular classes of $H_{2}(M,\partial
M;\mathbb{R})$. We just showed that $h$ is well-defined on the rational
regular points of $S$. Observe that property (3) of Theorem 8.3 implies that
$h$ is locally maximal on the rational regular points of $S$. That is, for any
neighborhood $U$ of a given rational regular point $b\in S$ that consists of
regular points, if $a$ is a rational point $U$ then $h(a)\leq h(b)$. We may
use this property to extend $h$ to the irrational regular points of $S$ and
hence to the rest of the regular classes of $H_{2}(M,\partial M;\mathbb{R})$.

Let $z$ be an irrational regular point of $S$. Then define $h(z)$ so that: For
any neighborhood $U$ of $z$ consisting of regular points, there exists a
neighborhood $U^{\prime}\subset U$ of $z$ for which $h(z)$ is the maximum of
$h$ among the rational points in $U^{\prime}$. Then we can see that $h$ is
locally maximal on the regular points of $S$.

Since the regular classes of $H_{2}(M,\partial M;\mathbb{R})-\\{0\\}$ are a
dense open set by Lemma 8.2, the regular points of $S$ also form a dense open
set of $S$. Thus we may extend $h$ to irregular points similarly to how we
extended $h$ to irrational regular points. Let $z$ be an irregular point of
$S$ and define $h(z)$ as follows: For any neighborhood $U$ of $z$, there is a
neighborhood $U^{\prime}\subset U$ of $z$ for which $h(z)$ is the maximum of
$h$ among the regular points in $U^{\prime}$. As before, we similarly see that
$h$ is locally maximal on $S$.

Consequently, with $h$ defined on $S$, we may extend $h$ to all of
$H_{2}(M,\partial M;\mathbb{R})-\\{0\\}$ so that it is constant on rays. If
$0$ is a regular class, then $h(M,\gamma,0)$ is defined and
$h(M,\gamma,\xi)\leq h(M,\gamma,0)-2d$ for any regular class $\xi$ by Theorem
6.5. Thus it follows that $h(0)\geq h(\xi)$ for any class $\xi\in
H_{2}(M,\partial M;\mathbb{R})$ since $d\geq 0$. If $0$ is not a regular
class, then we may just define $h(0)$ to be the maximum of $h(\xi)$ among the
regular classes $\xi$. By Theorem 8.3(1) and our extension of $h$, this
maximum exists. Hence, in either case, $h$ is bounded by $h(0)$. ∎

######  Remark 8.5.

It may happen that $h(M,\gamma,\xi)\neq h(M,\gamma,-\xi)$. Flipping the
orientation of a decomposing surface $F$ may significantly change the
resulting suture structure of the decomposed manifold.

######  Remark 8.6.

As shown in the proof of Theorem 8.4, when $0$ is a regular class $h(\xi)$ is
bounded above by the linear handle number of $(M,\gamma)$. Consequently, the
handle number function of a product sutured manifold is the constant function
$h=0$.

##  9\. Further remarks and questions.

###  9.1. Regarding handle numbers and faces of the Thurston norm

######  Remark 9.1.

For integral classes $\beta\in H_{2}(M,\partial M;\mathbb{Z})$, we have
$h(\beta)=0$ precisely when there is a surface representing $\beta$ that
decomposes $(M,\gamma)$ into a product sutured manifold. In light of Theorems
8.3 and 8.4, one may regard any class $\beta\in H_{2}(M,\partial
M;\mathbb{R})$ with $h(\beta)=0$ as a fibered class with respect to the
sutured manifold structure $(M,\gamma)$.

Indeed, if $\partial M$ is a union of tori and $\gamma$ has no annular
sutures, then for $\beta\neq 0$ we have $h(\beta)=0$ if and only if $\beta$ is
a fibered class in the traditional sense (ie. $\beta$ lies in the cone over a
fibered face of the Thurston norm). When $\gamma$ has annular sutures we will
have $h(\beta)>0$ for such a fibered class if $\partial\beta$ has the same
slope as an annular suture in some component of $\partial M$.

######  Question 9.2.

Let $(M,\gamma_{+})$ be a sutured manifold where $\partial M$ is a union of
tori and $R_{+}(\gamma_{+})=\partial M$. Is the handle number constant on any
face of the Thurston norm ball?

######  Remark 9.3.

If $M$ has tunnel number $1$, then this indeed is the case since the handle
number of any regular class is bounded above by $2$ as discussed in
Observation 7.1. Since the handle number is $0$ exactly on the fibered faces
(cf. Remark 9.1), it must be $2$ elsewhere. Hence the handle number is
constant on any face.

Exteriors of two-bridge links provide a class of such examples. Following
[EMnMGRLRV21, Theorem 28], one may determine all the fibered faces of the
Thurston norm of a two-bridge link exterior and hence the handle number
function. Note that some two-bridge links, such as $[0;4,4,-2]$, have no
fibered class, so their handle number functions are the constant function
$h=2$.

######  Remark 9.4.

Let $(M,\gamma)$ be a sutured manifold where $\partial M$ is a union of tori
and $R_{+}(\gamma)$ is a union of essential annuli. In particular each
component of $\partial M$ has some annular sutures of some slope. Then the
handle number function of $(M,\gamma)$ need not be constant on faces of the
Thurston norm.

###  9.2. Handle complexities of 2nd homology classes

The Thurston norm $x(\xi)$ and sutured characteristic $\chi_{(M,\gamma)}(\xi)$
measure the complexity of any class $\xi\in H_{2}(M,\partial M;\mathbb{Z})$ of
the sutured manifold $(M,\gamma)$ by minimizing the complexity $x$ or
$\chi_{(M,\gamma)}$ among surfaces representing $\xi$. (Recall that for an
oriented surface $F$, $x(F)=\chi_{-}(F)$ which is the sum of $-\chi(F_{0})$
among the components $F_{0}$ of $F$ which are neither spheres nor disks.)
Restricting to regular classes, we may create variants $x^{h}$ and
$\chi_{(M,\gamma)}^{h}$ of these, the handle characteristic and handle sutured
characteristic respectively, by minimizing these complexities among surfaces
representing a regular class $\xi$ that also realize $h(M,\gamma,\xi)$.

######  Question 9.5.

Does there always exist a surface $F$ that realizes both $x^{h}(\xi)$ and
$\chi_{(M,\gamma)}^{h}(\xi)$?

These definitions immediately imply that $x(\xi)\leq x^{h}(\xi)$ and
$\chi_{(M,\gamma)}(\xi)\leq\chi^{h}_{(M,\gamma)}(\xi)$. Lemma 6.2 implies that
$x(\xi)\leq\chi_{(M,\gamma)}(\xi)$ and
$x^{h}(\xi)\leq\chi_{(M,\gamma)}^{h}(\xi)$. Together, we have:

${{\chi_{(M,\gamma)}(\xi)}}$${{\chi_{(M,\gamma)}^{h}(\xi)}}$${{x(\xi)}}$${{x^{h}(\xi)}}$

$\scriptstyle\leq$

$\scriptstyle\leq$

$\scriptstyle{\leq}$$\scriptstyle{\leq}$

For a knot exterior $M=S^{3}\backslash\mathcal{N}(K)$ we may take a sutured
manifold structure $\gamma_{+}$ with $\partial M=R_{+}$. Then
$x(F)=\chi_{(M,\gamma_{+})}(F)-1=2g(F)-1$ for any Seifert surface $F$. So for
the Seifert class $\xi=[F]$, we have
$x(\xi)=\chi_{(M,\gamma_{+})}(\xi)-1=2g(K)-1$. It further follows that
$x^{h}(\xi)=\chi^{h}_{(M,\gamma_{+})}(\xi)-1$. Hence, in this situation,
Conjecture 7.12 that $g(K)=g_{MN}(K)$ asks if either of the horizontal
inequalities above — and hence both — are equalities. We may extend Conjecture
7.12 to other sutured manifolds.

######  Question 9.6.

For a non-trivial regular class $\xi\in H_{2}(M,\partial M;\mathbb{Z})$ of a
sutured manifold $(M,\gamma)$:

  * •

Does $\chi_{(M,\gamma)}(\xi)=\chi^{h}_{(M,\gamma)}(\xi)$?

  * •

Does $x(\xi)=x^{h}(\xi)$?

While $x$ extends to a (pseudo)norm on $H_{2}(M,\partial M;\mathbb{R})$, it is
not clear that $x^{h}$ behaves so well.

######  Question 9.7.

Does the triangle inequality hold for $x^{h}$? That is, for non-trivial
regular integral classes $\alpha$ and $\beta$ such that $\alpha+\beta$ is
regular, do we have

$x^{h}(\alpha+\beta)\leq x^{h}(\alpha)+x^{h}(\beta)?$

Recall that we develop the extension of $h(M,\gamma,\xi)$ to non-trivial
regular classes in $H_{2}(M,\partial M;\mathbb{R})$ and then complete it to
all classes in Section 8. So if there are situations where the triangle
inequality holds for $x^{h}$, does it extend to a (pseudo)norm?

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