# Relative Khovanov-Jacobsson Classes

Isaac Sundberg & Jonah Swann

###### Abstract.

To a smooth, compact, oriented, properly-embedded surface in the $4$-ball, we
define an invariant of its boundary-preserving isotopy class from the Khovanov
homology of its boundary link. Previous work showed that when the boundary
link is empty, this invariant is determined by the genus of the surface. We
show that this relative invariant: can obstruct sliceness of knots; detects a
pair of slices for $9_{46}$; is not hindered by detecting connected sums with
knotted $2$-spheres.

##  1\. Introduction

Conjectured within the framework of [Kho00], which defined what would become
known as Khovanov homology, was an application of the map on Khovanov homology
induced by an oriented link cobordism. More precisely, to an oriented link
cobordisms $\Sigma\colon L_{0}\to L_{1}$, Khovanov constructed a
diagrammatically-defined homomorphism
$\textnormal{Kh}(\Sigma)\colon\textnormal{Kh}(L_{0})\to\textnormal{Kh}(L_{1})$
and conjectured that, up to sign, it is invariant under isotopy of $\Sigma$
rel $\partial\Sigma$. Specifically, Khovanov noted that if this conjecture
held, then the following application might warrant further study: for a closed
surface $S\colon\varnothing\to\varnothing$, the induced map is an endomorphism
$\textnormal{Kh}(S)\colon\mathds{Z}\to\mathds{Z}$, whereby the integer
$n_{S}:=|\textnormal{Kh}(S)(1)|\in\mathds{Z}$ determining $\textnormal{Kh}(S)$
is an invariant of the ambient isotopy class of $S$. The conjecture was
eventually proven by Jacobsson [Jac04], and then later in [BN05] and [Kho06]
through independent methods. The integer $n_{S}$ became known as the Khovanov-
Jacobsson number of the surface $S$, and calculations of $n_{S}$ for certain
families of surface knots were established [CSS04]. In the end, Rasmussen
[Ras05] and Tanaka [Tan05] independently proved Khovanov-Jacobsson numbers are
trivially determined by the genus of $S$.

In this paper, we define a relative version of Khovanov-Jacobsson numbers
associated to link cobordisms $\Sigma\colon\varnothing\to L$. Namely, the
relative Khovanov-Jacobsson class of $\Sigma$ is the class

$\textnormal{KJ}_{\Sigma}=|\textnormal{Kh}(\Sigma)(1)|\in\textnormal{Kh}(L).$

It is an invariant of the isotopy class of $\Sigma$ rel $\partial\Sigma$. When
$L=\varnothing$, we use absolute in place of relative. A definition of this
invariant, and a discussion of its diagram dependence, is given in Section 3
.

The remainder of the paper studies and implements properties of relative
Khovanov-Jacobsson classes. In Section 4 , we characterize the relative
classes associated to surfaces bounding the unlink and surfaces obtained by
Seifert’s algorithm. In Section 5 , we show these classes can obstruct
sliceness of knots. We illustrate this on a class of $3$-stranded pretzel
knots.

The paper concludes by discussing boundary-preserving isotopy classes of
surfaces properly-embedded in the $4$-ball. Such classes have gained recent
attention in [JZ20], [CP20], and [MP20], where a refined characterization has
been suggested. Connect summing a surface with a knotted $2$-sphere, or
locally knotting a surface, generally changes the boundary-preserving isotopy
class of the surface. By considering surfaces up to both boundary-preserving
isotopy and local knotting, we create a more nuanced distinction between
surfaces, better capturing the boundary-preserving isotopy being established.
In Section 6 , we show that relative Khovanov-Jacobsson classes do not detect
local knottedness, implying they are an invariant of these refined isotopy
classes of surfaces. We conclude by showing relative Khvanov-Jacobsson classes
detect a pair of slices for $9_{46}$ and generalize this to arbitrary connect
sums of $9_{46}$.

##  2\. Background on Khovanov Homology

In this section, we provide the notation and jargon we use when discussing the
Khovanov functor. We assume familiarity with the construction of the Khovanov
chain complex, see [Kho00] or [BN02] among many others. Sections 2.2 is based
on the terminology from [Ell10], and Section 2.3 is based on the work from
[Jac04].

###  2.1. Khovanov Homology

To each diagram $D$ of an oriented link $L$ and each enumeration of its
crossings, we associate a bigraded chain complex
$\mathcal{C}\textnormal{Kh}(D)$ called the Khovanov chain complex of $D$. The
homology $\textnormal{Kh}(D)$ is called the Khovanov homology of $D$. This
definition is independent, up to chain homotopy equivalence, of both the
chosen enumeration of crossings in $D$ and the chosen diagram $D$. In
particular, associated to each Reidemeister move is a pseudoisomorphism
between Khovanov chain complexes, discussed briefly in Section 2.3 . Thus,
the chain homotopy class $\mathcal{C}\textnormal{Kh}(L)$ of
$\mathcal{C}\textnormal{Kh}(D)$ is an invariant of the link $L$ called the
Khovanov chain complex of $L$; the isomorphism class $\textnormal{Kh}(L)$ of
$\textnormal{Kh}(D)$ is called the Khovanov homology of $L$.

###  2.2. Elements of the Khovanov chain complex

Here, we provide language used on elements of the Khovanov chain complex. Let
$D$ be a diagram for an oriented link $L$ having $n$ crossings, of which there
are $n_{+}$ positive crossings and $n_{-}$ negative crossings. Enumerate the
crossings of $D$.

A crossing  of $D$ can be smoothed as either a $0$-smoothing or a
$1$-smoothing .

A state $\sigma$ of $D$ is a planar $1$-manifold obtained by replacing each
crossing in $D$ with either a $0$-smoothing or $1$-smoothing. A diagram with
$n$-many crossings will have $2^{n}$-many states. A state can be represented
as a binary sequence $\sigma=(v_{1},\dots,v_{n})$, where the $i$th crossing is
$v_{i}$-smoothed. The height of a smoothing is $|\sigma|=\sum v_{i}$. The
state whose crossings are all $0$-smoothed (or $1$-smoothed) is called the
$0$-state (or $1$-state).

A trace is an indicator placed on a smoothed crossing to record the type of
smoothing: a $0$-trace is a red dot-dashed band indicating a $0$-smoothing; a
$1$-trace is a blue dotted band indicating a $1$-smoothing. A trace state is a
state whose smoothings are labeled with the appropriate traces.

An enhanced state $\alpha_{\sigma}$ is a state $\sigma$ whose components are
labeled with either a $1$ or an $x$. A state with $m$ components has $2^{m}$
associated enhanced states. We record the number $v_{+}(\alpha)$ of $1$-labels
and the number $v_{-}(\alpha)$ of $x$-labels. The enhanced state whose loops
are all $1$-labeled is called the all $1$-label and is denoted $\sigma_{1}$;
the enhanced state whose loops are all $x$-labeled is called the all $x$-label
and is denoted $\sigma_{x}$. An enhanced trace state is both an enhanced state
and a trace state.

We define a bigrading $\mathcal{C}\textnormal{Kh}^{h,q}(D)$ on enhanced states
having homological and quantum gradings:

$\displaystyle h(\alpha_{\sigma})$ $\displaystyle=|\sigma|-n_{-}$
$\displaystyle q(\alpha_{\sigma})$
$\displaystyle=v_{+}(\alpha_{\sigma})-v_{-}(\alpha_{\sigma})+h(\alpha_{\sigma})+n_{+}-n_{-}$

Each chain group $\mathcal{C}\textnormal{Kh}^{h,q}(D)$ is generated by the
possible enhanced states from this bigrading 111Formally, the generators of
$\mathcal{C}\textnormal{Kh}(D)$ are algebraic objects which can be viewed
pictorially as enhanced states. .

Figure 1. From left to right: the figure $8$ knot; a state $\sigma$ of the
figure $8$ knot; the trace state of $\sigma$; one of the $2^{3}$ enhanced
trace states on $\sigma$; the trace state graph $\Gamma_{\sigma}$.

Associated to a trace state $\sigma$ is a trace state graph $\Gamma_{\sigma}$
whose vertices are the components of $\sigma$ and whose edges are the traces.
When illustrating $\Gamma_{\sigma}$, edges can be drawn using bands that
mirror the trace they represent. Moreover, we often show $\sigma$ as a shadow
of the graph, as in Figure 1 . We interchangeably refer to the loops (or
traces) in $\sigma$ and the vertices (or edges) in $\Gamma_{\sigma}$. We will
have occasion to refer to certain subgraphs of the state graph. Let
$\Gamma_{0}$ denote the subgraph whose edges consist of all $0$-traces, and
let $\Gamma_{1}$ be the subgraph of $1$-traces.

One chain element that we encounter regularly in this paper is called a
$pqr$-chain, which we have illustrated in Figure 2 . The left side of the
figure shows the shorthand we use to describe such chains. In this shorthand,
a loop $\gamma$ labeled with a letter of the alphabet (usually beginning with
$p$ and proceeding alphabetically) corresponds to a summand in the $pqr$-chain
where $\gamma$ is $1$-labeled and all other labeled loops are $x$-labeled. If
the letter on $\gamma$ is capitalized, the summand has a negative sign. A loop
that is unlabeled in the shorthand is $1$-labeled in each summand. Using
$pqr$-chains significantly reduces the number of enhanced states that must be
illustrated in certain chain elements. We will make particularly good use of
this in the upcoming sections.

Figure 2. A $pqr$-chain on the all $0$-state of the figure $8$ knot .

###  2.3. Maps on Khovanov homology

Here, we describe the map on Khovanov homology induced by a link cobordism. A
link cobordism $\Sigma\colon L_{0}\to L_{1}$ is a smooth, compact, oriented,
properly-embedded surface $\Sigma\subset\mathds{R}^{3}\times[0,1]$ whose
boundary is a pair of oriented links
$L_{i}=\Sigma\cap(\mathds{R}^{3}\times\\{i\\})$. A generic 222A link cobordism
$\Sigma\colon L_{0}\to L_{1}$ is generic if, with respect to the $[0,1]$
factor of $\mathds{R}^{3}\times[0,1]$, it restricts to a Morse function with
distinct critical values. When generic, each level set
$L_{t}=\Sigma\cap(\mathds{R}^{3}\times\\{t\\})$ is a link except at finitely
many critical levels $t_{1},\dots,t_{n}$, where it contains either a
transverse double point or an isolated point. In this paper, we assume all
link cobordisms are generic. link cobordism with surface diagram 333A surface
diagram $S\colon D_{0}\to D_{1}$ of a generic link cobordism $\Sigma\colon
L_{0}\to L_{1}$ is the image $S\subset\mathds{R}^{2}\times I$ of $\Sigma$
under a generic projection $(p\times\text{id})\colon\mathds{R}^{3}\times
I\to\mathds{R}^{2}\times I$. Under these conditions, each
$D_{i}=S\cap(\mathds{R}^{2}\times\\{i\\})$ is a diagram of the boundary link
$L_{i}$ except at finitely many critical levels $t_{1},\dots,t_{n}$,
corresponding to the critical levels of $\Sigma$. $S\colon D_{0}\to D_{1}$
induces a movie 444A movie $D_{t}$ associated to a surface diagram $S\colon
D_{0}\to D_{1}$ of a generic link cobordism $\Sigma\colon L_{0}\to L_{1}$ is a
collection of frames $D_{t}=S\cap\mathds{R}^{2}\times\\{t\\}$. In a
sufficiently small neighborhood $[t_{i}-\varepsilon(i),t_{i}+\varepsilon(i)]$
of each critical level $t_{i}$ of $S$, the diagrams $D_{t_{i}-\varepsilon(i)}$
and $D_{t_{i}+\varepsilon(i)}$ are related by either a Morse move or
Reidemeister move. In each interval $(t_{i},t_{i+1})$, the movie describes an
isotopy between $D_{t_{i}+\varepsilon(i)}$ and $D_{t_{i+1}-\varepsilon(i+1)}$.
Thus, a movie can be realized as a finite sequence of isotopies, Morse moves,
and Reidemeister moves applied to $D_{0}$ and ending with $D_{1}$. $D_{t}$.
This movie can be decomposed into a finite sequence of link diagrams, with
each successive pair related by an isotopy, Morse move, or Reidemeister move.
These finite sequences of diagrams provide a way to explicitly express a link
cobordism.

For diagrams $D_{0}$ and $D_{1}$ related by a Morse or Reidemeister move,
there is a bigraded chain map
$\mathcal{C}\textnormal{Kh}(D_{0})\to\mathcal{C}\textnormal{Kh}(D_{1})$. The
Reidemeister induced maps are chain equivalences. Explicit statements of these
chain maps exist throughout the literature; we will not give them here. The
Morse induced maps are those used in constructing the Khovanov chain complex,
and can be found in most resources presenting this construction. Explicit
definitions of the Reidemeister moves are harder to come by. In [Ell09], chain
maps for the first and second Reidemeister moves are defined on enhanced
states. In [BN05], all but one Reidemeister III move is given using a version
of Khovanov homology defined categorically with cobordisms.

To define the maps on Khovanov homology induced by a link cobordism
$\Sigma\colon L_{0}\to L_{1}$, we use the following process. Choose a surface
diagram $S\colon D_{0}\to D_{1}$ for $\Sigma$ and express it as a movie
$D_{t}$. Shorten the movie to a finite sequence of diagrams, with each
successive pair of diagrams related by an isotopy, Morse move, or Reidemeister
move. Each of these relations between pairs of diagrams corresponds to a chain
map. Sequentially compose the chain maps according to the sequence of
diagrams. The result is a bigraded chain map on the Khovanov chain complex

$\mathcal{C}\textnormal{Kh}(S)\colon\mathcal{C}\textnormal{Kh}^{i,j}(D_{0})\to\mathcal{C}\textnormal{Kh}^{i,j+\chi(\Sigma)}(D_{1})$

inducing a bigraded homomorphism $\textnormal{Kh}(S)$ on homology. This chain
map is invariant, up to sign and chain homotopy, under isotopy of $\Sigma$ rel
$\partial\Sigma$, as in the following theorem.

######  Theorem 2.2.

[Jac04] If $S,S^{\prime}\colon D_{0}\to D_{1}$ are surface diagrams for link
cobordisms $\Sigma,\Sigma^{\prime}\colon L_{0}\to L_{1}$ that are isotopic
relative to their common boundary, then
$\mathcal{C}\textnormal{Kh}(S)\sim\pm\mathcal{C}\textnormal{Kh}(S^{\prime})$.

Thus, the induced homomorphism $\textnormal{Kh}(S)$ is an up-to-sign invariant
of the boundary-preserving isotopy class of $\Sigma$. Throughout this paper,
we will use the common shorthand
$\mathcal{C}\textnormal{Kh}(\Sigma):=\mathcal{C}\textnormal{Kh}(S)$ and
$\textnormal{Kh}(\Sigma):=\textnormal{Kh}(S)$. The reader is warned that this
can be misleading, as it does not specify the surface diagram with which the
domain and codomain are defined. When one is not specified, assume we are
working with a surface diagram, whereby the (co)domain of either map are
defined by the associated link diagrams.

######  Remark 2.3.

The induced map is functorial with respect to composition 555A pair of link
cobordisms $\Sigma\colon L_{0}\to L_{1}$ and $\Sigma^{\prime}\colon-L_{1}\to
L_{2}$, can be composed by scaling $\Sigma$ into
$\mathds{R}^{2}\times[0,\frac{1}{2}]$ and $\Sigma^{\prime}$ into
$\mathds{R}^{2}\times[\frac{1}{2},1]$, forming the link cobordism
$\Sigma^{\prime}\circ\Sigma\colon L_{0}\to L_{2}$. of link cobordisms: for
link cobordisms $\Sigma\colon L_{0}\to L_{1}$ and $\Sigma^{\prime}\colon-
L_{1}\to L_{2}$, the induced Khovanov chain maps satisfy

$\mathcal{C}\textnormal{Kh}(\Sigma^{\prime})\circ\mathcal{C}\textnormal{Kh}(\Sigma)=\mathcal{C}\textnormal{Kh}(\Sigma^{\prime}\circ\Sigma).$

Note that this composition makes sense, as
$\mathcal{C}\textnormal{Kh}(L_{1})=\mathcal{C}\textnormal{Kh}(-L_{1})$.
Conversely, a link cobordism can be decomposed as a composition of a pair of
link cobordisms by cutting along some non-critical level of the cobordism. In
such a case, the induced map also decomposes in the expected manner.

##  3\. Relative Khovanov-Jacobsson Classes

In this section, we define the relative Khovanov-Jacobsson cycle and class
associated to a link cobordism $\varnothing\to L$. We discuss the dependence
of each on the chosen diagram of the boundary link.

###  3.1. Definition

In the previous section, we showed how a link cobordism
$\Sigma\colon\varnothing\to L$ with surface diagram $S\colon\varnothing\to D$
induces a chain map
$\mathcal{C}\textnormal{Kh}(S)\colon\mathcal{C}\textnormal{Kh}(\varnothing)\to\mathcal{C}\textnormal{Kh}(D)$
with induced homomorphism
$\textnormal{Kh}(S)\colon\textnormal{Kh}(\varnothing)\to\textnormal{Kh}(D)$.
Since both $\mathcal{C}\textnormal{Kh}(\varnothing)$ and
$\textnormal{Kh}(\varnothing)$ are infinite cyclic, or more generally some
commutative ring with unity $1$, this map is determined by the image of $1$.
Theorem 2.2 then implies

$\textnormal{KJ}_{\Sigma,D}:=|\textnormal{Kh}(\Sigma)(1)|$

is an invariant of the boundary-preserving isotopy class of $\Sigma$. We call
this element the relative Khovanov-Jacobsson class of $\Sigma$ with respect to
the diagram $D$.

In practice, the chain element $\mathcal{C}\textnormal{Kh}(\Sigma)(1)$ is more
valuable than $\textnormal{KJ}_{\Sigma,D}$, as it encodes all the necessary
information for both maps $\mathcal{C}\textnormal{Kh}(\Sigma)$ and
$\textnormal{Kh}(\Sigma)$. For this reason, the element

$\mathcal{C}\textnormal{KJ}_{\Sigma,D}:=\mathcal{C}\textnormal{Kh}(\Sigma)(1)$

is called the relative Khovanov-Jacobsson cycle, and, up to sign, it
represents $\textnormal{KJ}_{\Sigma,D}$.

We should verify $\mathcal{C}\textnormal{KJ}_{\Sigma,D}$ is a cycle. The chain
complex $\mathcal{C}\textnormal{Kh}(\varnothing)$ is supported in the
$(0,0)$-grading, so we must have $d(1)=0$. Because
$\mathcal{C}\textnormal{Kh}_{\Sigma}$ is a chain map, it follows that

$d(\mathcal{C}\textnormal{KJ}_{\Sigma,D})=d(\mathcal{C}\textnormal{Kh}(\Sigma)(1))=\mathcal{C}\textnormal{Kh}(\Sigma)(d(1))=\mathcal{C}\textnormal{Kh}(\Sigma)(0)=0.$

So $\mathcal{C}\textnormal{KJ}_{\Sigma,D}$ is a cycle in
$\mathcal{C}\textnormal{Kh}(D)$, representing a homology class in
$\textnormal{Kh}(D)$.

As said in the introduction, the case where $L=\varnothing$ has been fully
established. In this case, we call $\textnormal{KJ}_{\Sigma,\varnothing}$ the
Khovanov-Jacobsson number of $\Sigma$. In light of this paper, one might call
it an absolute Khovanov-Jacobsson class. Initial observation shows that,
because $\mathcal{C}\textnormal{Kh}(\varnothing)$ is supported in the
$(0,0)$-grading and $\mathcal{C}\textnormal{Kh}(\Sigma)$ is a
$(0,\chi(\Sigma))$-graded map, we have
$\textnormal{KJ}_{\Sigma,\varnothing}=0$ for surfaces with $g(\Sigma)\neq 1$.
The case where $g(\Sigma)=1$ is harder, but the Khovanov-Jacobsson numbers
were established:

######  Theorem 3.1.

[Ras05] [Tan05] A link cobordism $\Sigma\colon\varnothing\to\varnothing$ with
genus $g(\Sigma)=1$ has absolute Khovanov-Jacobsson class
$\textnormal{KJ}_{\Sigma,\varnothing}=2$.

######  Remark 3.2.

Relative Khovanov-Jacobsson classes bear some resemblance to the invariant
from knot Floer homology defined in [JM16]. It is unclear if the two
invariants are related (for example, by the Dowlin spectral sequence).

###  3.2. Diagram Dependence

The relative Khovanov-Jacobsson class of a link cobordism
$\Sigma\colon\varnothing\to\\!L$ depends on the chosen surface diagram
$S\colon\varnothing\to D$. More specifically, it depends on the chosen link
diagram: a diagram for the boundary link induces a surface diagram for the
link cobordism because both are defined by the same projection. This
dependence is straightforward: the diagram specifies the chain group and
homology group to which $\mathcal{C}\textnormal{KJ}_{\Sigma,D}$ and
$\textnormal{KJ}_{\Sigma,D}$ respectively belong. When we change the surface
diagram, these groups change by a Reidemeister induced equivalence. To measure
the diagram dependence of relative Khovanov-Jacobsson classes, one could ask
whether they are preserved by the Reidemeister induced equivalences. In this
way, one could say that relative Khovanov-Jacobsson classes are link cobordism
invariants, being invariant of the chosen surface diagram. We prove this fact
in Proposition 3.4, however, we must first make a definition.

######  Remark 3.3.

Because we are studying the boundary-preserving isotopy class of $\Sigma$, we
emphasize that this paper only considers the link $L$, and not its isotopy
class. Moreover, we emphasize that the permissible chain homotopies defining
$\mathcal{C}\textnormal{Kh}(L)$ arise from a sequence of link specific
Reidemeister moves (made precise below), and not by any sequence of
Reidemeister moves.

A diagram for a link $L$ is the image of some projection
$p\colon\mathds{R}^{3}\to\mathds{R}^{2}$ onto a codimension-one, linear
subspace. Projections $p,p^{\prime}\colon\mathds{R}^{3}\to\mathds{R}^{2}$
defining diagrams $D,D^{\prime}$ are related by a one-parameter family of
rotations $r_{t}\colon\mathds{R}^{3}\times I\to\mathds{R}^{3}$ with
$r_{0}=\text{id}$ taking one projection onto the other $p^{\prime}=p\circ
r_{1}$. A small perturbation of $r$ makes $p\circ r_{t}$ generic as a link
projection, whereby $(p\circ r_{t})(L)$ describes a sequence of link specific
Reidemeister moves from $D$ to $D^{\prime}$, meaning each Reidemeister move
relates a pair of diagrams specific to $L$. The chain map induced by these
Reidemeister moves, and the map it induces on homology, will be called a link
specific Reidemeister induced map.

Thus, $\mathcal{C}\textnormal{Kh}(L)$ denotes the chain homotopy class of
$\mathcal{C}\textnormal{Kh}(D)$ up to link specific Reidemeister induced chain
homotopy equivalences, and $\textnormal{Kh}(L)$ denotes the isomorphism class
of $\textnormal{Kh}(D)$ under link specific Reidemeister induced isomorphisms.

######  Proposition 3.4.

For a link cobordism $\Sigma\colon\varnothing\to L$ with surface diagrams
$S,S^{\prime}\colon\varnothing\to D,D^{\prime}$, this diagram commutes, up to
sign and up to homotopy, for link specific Reidemeister induced maps
$\varphi$.

${\mathcal{C}\textnormal{Kh}(D^{\prime})}$${\mathcal{C}\textnormal{Kh}(D)}$${\mathcal{C}\textnormal{Kh}(\varnothing)}$$\scriptstyle{\varphi}$$\scriptstyle{\mathcal{C}\textnormal{Kh}(S^{\prime})}$$\scriptstyle{\mathcal{C}\textnormal{Kh}(S)}$

Thus, the class $\mathcal{C}\textnormal{KJ}_{\Sigma,L}$ of relative Khovanov-
Jacobsson cycles $\mathcal{C}\textnormal{KJ}_{\Sigma,D}$ related by link
specific Reidemeister induced chain maps, and the class
$\textnormal{KJ}_{\Sigma,L}$ defined similarly, are link cobordism invariants.

Proof. The idea is to construct a pair of link cobordisms inducing
$\varphi\circ\mathcal{C}\textnormal{Kh}(S^{\prime})$ and
$\mathcal{C}\textnormal{Kh}(S)$ that are isotopic relative to $L$, whereby
Theorem 2.2  implies the diagram commutes in the desired manner. The tricky
part is producing isotopic cobordisms whose boundaries produce identical
diagrams with respect to the same projection.

Let $p,p^{\prime}\colon\mathds{R}^{3}\to\mathds{R}^{2}$ be the projections
defining the surface diagrams $S$ and $S^{\prime}$. Remark 3.3 gives a one-
parameter family of rotations $r_{t}$ with $r_{0}=\text{id}$ and
$p^{\prime}=p\circ r_{1}$ that induces a movie $(p\circ r_{t})(L)$ describing
a sequence of link specific Reidemeister moves from $D$ to $D^{\prime}$.
Consider the one-parameter family of link cobordisms describing the trace of
$L$ under this movie:

$A_{s}=r|_{L\times[0,1-s]}\colon r_{0}(L)\to r_{1-s}(L)$

The link cobordism $A_{0}^{-1}\colon r_{1}(L)\to L$ has surface diagram
$p(A_{0}^{-1})\colon D^{\prime}\to D$, so with respect to $p$, this link
cobordism $A_{0}^{-1}\circ r_{1}(\Sigma)$ induces the map
$\varphi\circ\mathcal{C}\textnormal{Kh}(S^{\prime})$. Since $\Sigma$ induces
$\mathcal{C}\textnormal{Kh}(S)$ with respect to $p$, it suffices to show
$A_{0}^{-1}\circ r_{1}(\Sigma)$ and $\Sigma$ are isotopic relative to $L$.
Indeed, $A_{s}^{-1}\circ r_{s}(\Sigma)$ describes a boundary-preserving
isotopy between these link cobordisms, as desired. ∎

In light of this proposition, the diagram defining a relative Khovanov-
Jacobsson cycle or class is either clear from context or is unlikely to
provide any resistance to an argument. Thus, we often omit it from our
notation, writing $\mathcal{C}\textnormal{KJ}_{\Sigma}$ and
$\textnormal{KJ}_{\Sigma}$ instead of $\mathcal{C}\textnormal{KJ}_{\Sigma,D}$
and $\textnormal{KJ}_{\Sigma,D}$.

##  4\. Calculating Relative Khovanov-Jacobsson Classes

In this section, we provide a general commentary on calculating relative
Khovanov-Jacobsson classes. We give two classifications: for surfaces bounding
the unlink and for surfaces obtained by Seifert’s algorithm. We discuss
techniques for determining nontriviality of enhanced state cycles.

###  4.1. Calculating Induced Maps

Calculating the relative Khovanov-Jacobsson class of a link cobordism
$\Sigma\colon\varnothing\to L$ depends entirely on calculating the cycle
$\mathcal{C}\textnormal{KJ}_{\Sigma}$, which in turn requires a thorough
understanding of the chain map $\mathcal{C}\textnormal{Kh}(\Sigma)$. As
mentioned in Section 2.3, the chain maps defining
$\mathcal{C}\textnormal{Kh}(\Sigma)$ can be found in numerous locations
throughout the literature. For an example calculation using these maps, see
Figure 10 at the end of the paper.

Although general calculations require a thorough understanding of
$\mathcal{C}\textnormal{Kh}(\Sigma)$, there are extreme cases that can be
understood from a theoretical standpoint, as in the following two theorems.

###  4.2. Calculations for Surfaces Bounding the Unlink

Here, we classify relative Khovanov-Jacobsson classes for surfaces bounding
the unlink, as in the following theorem.

######  Theorem 4.2.

Let $U$ be the $n$-component unlink and $D$ its crossingless diagram. The
relative Khovanov-Jacobsson class for a connected link cobordism
$\Sigma:\varnothing\to U$ is determined by its genus:

  * (a)

if $g(\Sigma)=0$, then $\mathcal{C}\textnormal{KJ}_{\Sigma,D}$ is, up to sign,
a $pqr$-chain on the components of $D$;

  * (b)

if $g(\Sigma)=1$, then $\mathcal{C}\textnormal{KJ}_{\Sigma,D}$ is, up to sign,
twice the all $x$-label $\sigma_{x}$;

  * (c)

if $g(\Sigma)\geqslant 2$, then $\mathcal{C}\textnormal{KJ}_{\Sigma,D}=0$.

Proof. The lemma follows from an analysis on the bigrading to which
$\mathcal{C}\textnormal{KJ}_{\Sigma}$ belongs. We first determine the support
of the chain complex $\mathcal{C}\textnormal{Kh}(D)$. As $D$ is a crossingless
diagram, the chain complex $\mathcal{C}\textnormal{Kh}(D)$ is supported in the
$0$-homological grading. To find the support of the quantum grading, we
consider the enhanced states of $D$, which generate the chain groups of
$\mathcal{C}\textnormal{Kh}(D)$. The quantum grading of an enhanced state
$\alpha$ for the crossingless diagram $D$ of $U$ is given by

(1) $q(\alpha)=v_{+}(\alpha)-v_{-}(\alpha)$

As $D$ is an $n$-component crossingless diagram, we also have

(2) $n=v_{+}(\alpha)+v_{-}(\alpha)$

Neither of $v_{\pm}(\alpha)$ can be negative, so it follows that the chain
group $\mathcal{C}\textnormal{Kh}(D)$ is supported in the $(0,j)$ grading for
$-n\leqslant j\leqslant n$.

To determine the exact bigrading for $\mathcal{C}\textnormal{KJ}_{\Sigma}$,
recall that the induced map $\mathcal{C}\textnormal{Kh}(\Sigma)$ is a
$(0,\chi(\Sigma))$ bigraded chain map, taking
$1\in\mathcal{C}\textnormal{Kh}^{0,0}(\varnothing)$ to a single quantum
grading, $q(\mathcal{C}\textnormal{Kh}(\Sigma)(1))=\chi(\Sigma)$. Thus,
$\mathcal{C}\textnormal{KJ}_{\Sigma}$ belongs to the chain group
$\mathcal{C}\textnormal{Kh}^{0,\chi(\Sigma)}(D)$, whereby it can be expressed
as a linear combination of enhanced states from the same bigrading:

$\hskip
150.0pt\mathcal{C}\textnormal{KJ}_{\Sigma}=\sum_{i=1}^{m}a_{i}\alpha_{i}\hskip
50.0pta_{i}\in\mathds{Z},\hskip 5.0ptq(\alpha_{i})=\chi(\Sigma).$

Furthermore, equations (1) and (2) combine with this quantum grading to show
$v_{+}(\alpha_{i})=1-g(\Sigma)$.

For $g(\Sigma)\geqslant 2$, the grading
$q(\mathcal{C}\textnormal{KJ}_{\Sigma})=2-2g(\Sigma)-n<-n$ is outside the
support, implying (c).

For $g(\Sigma)=1$, the enhanced states at this grading must satisfy
$v_{+}(\alpha_{i})=0$. So only the all $x$-label $\sigma_{x}$ is valid, and
$\mathcal{C}\textnormal{KJ}_{\Sigma}=a\sigma_{x}$ for some $a\in\mathds{Z}$.
To establish (b), we need only determine the value of the coefficient.
Consider the composition $C\circ\Sigma$, where $C$ is the surface that caps
off each boundary component of $\Sigma$ with a disk. Note that $C\circ\Sigma$
is a closed, connected surface with genus $1$, so by the characterization of
absolute Khovanov-Jacobsson classes (Theorem 3.1 ), we have

$\mathcal{C}\textnormal{Kh}(C)(\mathcal{C}\textnormal{KJ}_{\Sigma})=\mathcal{C}\textnormal{Kh}(C)(\mathcal{C}\textnormal{Kh}(\Sigma)(1))=\mathcal{C}\textnormal{Kh}(C\circ\Sigma)(1)=\mathcal{C}\textnormal{KJ}_{C\circ\Sigma}=\pm
2.$

Moreover, $C$ induces the map $\otimes_{n}\varepsilon$, where
$\varepsilon=\\{1\mapsto 0;x\mapsto 1\\}$, on the enhanced states of $D$, so

$\mathcal{C}\textnormal{Kh}(C)(\mathcal{C}\textnormal{KJ}_{\Sigma})=\mathcal{C}\textnormal{Kh}(C)(a\alpha_{-})=a.$

We conclude that $\mathcal{C}\textnormal{KJ}_{\Sigma}=\pm 2\alpha_{-}$,
establishing (b).

For $g(\Sigma)=0$, the only enhanced states at this grading are those
consisting of a single $1$-label. In order to conclude (a), we must show that
$a_{i}=\pm 1$ and that all the $a_{i}$ have the same sign. Enumerate the
components $U_{i}$ of $U$, and consistently carry over this enumeration to the
components $D_{i}$ of of the crossingless diagram $D$. Let $\alpha_{i}$ be the
enhanced state with $1$-labeled $D_{i}$ and $x$-labeled $D_{j\neq i}$.

To see that each $a_{i}=\pm 1$, consider $C_{j}\circ\Sigma$, where $C_{j}$ is
the surface that caps off $U_{j}$ with a punctured torus and the remaining
boundary component $U_{i\neq j}$ with a disk. Note $C_{j}\circ\Sigma$ is a
closed, connected surface with genus $1$, so by the characterization of
absolute Khovanov-Jacobsson classes,

$\mathcal{C}\textnormal{Kh}(C_{j})(\mathcal{C}\textnormal{KJ}_{\Sigma})=\mathcal{C}\textnormal{Kh}(C_{j})(\mathcal{C}\textnormal{Kh}(\Sigma)(1))=\mathcal{C}\textnormal{Kh}(C_{j}\circ\Sigma)(1)=\pm
2.$

Moreover, $C_{j}$ induces the map $\varepsilon$ (defined above) on each
$D_{i}$ and $\\{1\mapsto 2;x\mapsto 0\\}$ on $D_{j}$. It follows that
$\mathcal{C}\textnormal{Kh}(C_{j})(\alpha_{j})=2$ and
$\mathcal{C}\textnormal{Kh}(C_{i\neq j})(\alpha_{j})=0$, so applying $C_{j}$
to $\mathcal{C}\textnormal{KJ}_{\Sigma}$, we have

$\mathcal{C}\textnormal{Kh}(C_{j})(\mathcal{C}\textnormal{KJ}_{\Sigma})=\mathcal{C}\textnormal{Kh}(C_{j})\bigg{(}\sum_{i=1}^{m}a_{i}\alpha_{i}\bigg{)}=\sum_{i=1}^{m}a_{i}\mathcal{C}\textnormal{Kh}(C_{j})(\alpha_{i})=a_{j}\mathcal{C}\textnormal{Kh}(C_{j})(\alpha_{j})=2a_{j}$

So $a_{j}=\pm 1$, as desired.

Figure 3. From left to right: the surfaces $C_{j}$ and $C_{j,k}$.

To see each $a_{i}$ has the same sign, consider $C_{j,k}\circ\Sigma$, where
$C_{j,k}$ is the surface that caps off each $U_{i\neq j,k}$ with a disk and
connects the remaining two $U_{j}$ and $U_{k}$ with an anulus. A nearly
identical calculation shows $a_{j}+a_{k}=\pm 2$ for each $j\neq k$, which
holds only when $a_{j}=\pm 1$ and $a_{k}=\pm 1$ having the same sign. Thus,
$\mathcal{C}\textnormal{KJ}_{\Sigma}$ is the desired $pqr$-chain on $D$. ∎

######  Remark 4.3.

Theorem 4.2 assumes that $\Sigma$ is a connected surface. In the case that
there are multiple components, we may draw on the recent result of [GL20],
which shows that for split links, the induced map
$\mathcal{C}\textnormal{Kh}(\Sigma)$ is determined by the maps induced between
components of $\Sigma$, independent of their potential linking. Thus, this
theorem can be applied to each component of $\Sigma$ individually.

###  4.3. Calculations for Surfaces Obtained from Seifert’s Algorithm

Here, we classify the relative Khovanov-Jacobsson classes of surfaces obtained
by applying Seifert’s algorithm.

Recall that to a diagram $D$ of an oriented link, Seifert’s algorithm
associates an oriented, compact surface $\Sigma_{D}\subset S^{3}$ with
boundary $L$ by applying the following procedure: produce a smoothing
$\sigma_{D}$ of $D$ according to its orientation 666A diagram is smoothed
according to its orientation if each positive (negative) crossing is
$0$-smoothed ($1$-smoothed). , resulting in a collection of planar Seifert
circles, that can be oriented accordingly; each Seifert circle bounds an
appropriately oriented Seifert disk in the plane, and these disks can be
situated in $S^{3}$ so that an outermost disk is below any disk it contains;
each crossing in $D$ connects two Seifert disks and is realized as the
boundary of a twisted band between the disks. The collection of disks and
bands is an oriented surface $\Sigma_{D}$, called a Seifert surface. There is
a movie description of $\Sigma_{D}$ inherent to the structure of this
algorithm:

  * (1)

birth each Seifert disk in ascending order of their height (lowest, or
outermost, first);

  * (2)

for each negative crossing, perform a negative twist-and-glue move (Figure 4
) consisting of a single negative Reidemeister I and a single Morse saddle;

  * (3)

for each positive crossing, perform a positive twist-and-glue move (Figure 5
) consisting of a single positive Reidemeister I and a single Morse saddle.

Following this description, decompose the Seifert surface
$\Sigma_{D}=\Sigma_{3}\circ\Sigma_{2}\circ\Sigma_{1}$, where each $\Sigma_{i}$
is the surface corresponding to the $i$-th scene of the movie listed above.

Figure 4. The negative twist-and-glue move, along with its induced chain map.
Figure 5. The positive twist-and-glue move, along with its induced chain map.

######  Remark 4.4.

The cycle $\mathcal{C}\textnormal{KJ}_{\Sigma_{D}}$ produced by the above
movie description of $\Sigma_{D}$ is fairly tame, with the decomposition
$\Sigma_{D}=\Sigma_{3}\circ\Sigma_{2}\circ\Sigma_{1}$ allowing us to take this
calculation in stages. First, note that
$\mathcal{C}\textnormal{Kh}(\Sigma_{1})(1)$ is the all-$1$ label $\sigma_{1}$
on the orientation-induced smoothing $\sigma_{D}$. Second, each negative
twist-and-glue move induces the identity map, so
$\mathcal{C}\textnormal{Kh}(\Sigma_{2}\circ\Sigma_{1})(1)$ is again the
all-$1$ label $\sigma_{1}$ on $\sigma_{D}$. Third, each positive twist-and-
glue move does not change the underlying smoothing, but instead affects only
the labels. We conclude that

$\mathcal{C}\textnormal{KJ}_{\Sigma_{D}}=\mathcal{C}\textnormal{Kh}(\Sigma_{3}\circ\Sigma_{2}\circ\Sigma_{1})(1)=\mathcal{C}\textnormal{Kh}(\Sigma_{3})\big{(}\mathcal{C}\textnormal{Kh}(\Sigma_{2}\circ\Sigma_{1})(1)\big{)}=\mathcal{C}\textnormal{Kh}(\Sigma_{3})(\sigma_{1})$

is a sum of enhanced states on the smoothing $\sigma_{D}$, with labels
determined by the positive twist-and-glue moves.

######  Remark 4.5.

The positive crossings are all $0$-smoothed, so each positive twist-and-glue
move occurs on an edge of $\Gamma_{0}$ (see Section 2.2 ), whereby, the map
$\mathcal{C}\textnormal{Kh}(\Sigma_{3})$ is described by how it acts on the
labels of vertices in $\Gamma_{0}$. The vertices of the state graph
$\Gamma_{\sigma_{D}}$ that are not in the subgraph $\Gamma_{0}$ are
$1$-labeled, and are not affected by the positive twist-and-glue moves. The
following theorem shows that the homotopy type of the subgraph $\Gamma_{0}$
determines $\mathcal{C}\textnormal{KJ}_{\Sigma_{D}}$.

######  Theorem 4.6.

The relative Khovanov-Jacobsson cycle
$\mathcal{C}\textnormal{KJ}_{\Sigma_{D}}$ of a Seifert surface $\Sigma_{D}$ is
determined by the first Betti number of the subgraph $\Gamma_{0}$ of the state
graph $\Gamma_{\sigma_{D}}$ corresponding to the orientation-induced smoothing
$\sigma_{D}$ of the chosen diagram $D$. Explicitly, for connected $\Gamma_{0}$
we have:

  * (a)

if $b^{1}(\Gamma_{0})=0$, then $\mathcal{C}\textnormal{KJ}_{\Sigma_{D}}$ is a
$pqr$-chain on the vertices of $\Gamma_{0}$ (the sign of each summand in the
$pqr$-chain is determined by the parity of the distance of its unique
$1$-labeled vertex from a predetermined vertex $v_{0}$ of $\Gamma_{0}$).

  * (b)

if $b^{1}(\Gamma_{0})=1$, then $\mathcal{C}\textnormal{KJ}_{\Sigma_{D}}$ is
twice the enhanced state having $x$-labels on the vertices of $\Gamma_{0}$;

  * (c)

if $b^{1}(\Gamma_{0})\geqslant 2$, then
$\mathcal{C}\textnormal{KJ}_{\Sigma_{D}}=0$.

Proof. Suppose first that $b^{1}(\Gamma_{0})=0$. Choose a vertex $v_{0}$ in
$\Gamma_{0}$. Partially order the edge set
$E_{0}=\\{e_{1},e_{2},\dots,e_{k}\\}$ by their distance from $v_{0}$, and
situate $\Sigma_{3}$ so the positive twist-and-glue moves are done in this
order. The edge $e_{1}$ connects $v_{0}$ to some $v_{1}$, both of which are
$1$-labeled. The positive twist-and-glue move associated to $e_{1}$ produces a
$pqr$-chain on these vertices. We claim that successively applying the
remaining maps induced by the edges $e_{2},\dots,e_{k}$ will produce a
$pqr$-chain on $\Gamma_{0}$. Inducting on the edge set, suppose the edge
$e_{i}$ produces a $pqr$-chain on the vertices $v_{0},\dots,v_{i}$ incident to
one of $e_{1},\dots,e_{i}$. The edge $e_{i+1}$ connects one of these vertices,
say $v_{j}$, to a new, $1$-labeled vertex $v_{i+1}$. To see that the map
induced by $e_{i+1}$ produces a $pqr$-chain on the vertices
$v_{1},\dots,v_{i},v_{i+1}$, note that when $v_{j}$ is $x$-labeled, the
positive twist-and-glue map incorporates $v_{i}$ into the new $pqr$-chain as
another $x$-labeled loop, and when $v_{j}$ is $1$-labeled, the map produces
the pair of unique summands in the new $pqr$-chain with $1$-labels on either
$v_{i+1}$ or $v_{j}$. The sign of the unique summand with $1$-labeled
$v_{i+1}$ is determined by the sign of $v_{j}$: the sign indicated in the
positive twist-and-glue map (Figure 5 ) is either inherited by $v_{i+1}$
(when $v_{j}$ is positive) or is doubly-negated (when $v_{j}$ is negative).
Consequently, the sign is determined by the parity of the distance of
$v_{i+1}$ from $v_{0}$. It is negative when the distance is odd and positive
when even. We conclude that $\mathcal{C}\textnormal{KJ}_{\Sigma_{D}}$ has the
desired description.

Suppose $b^{1}(\Gamma_{0})=1$. Choose a maximal tree $G<\Gamma_{0}$ and let
$e$ be the unique remaining edge. Situate $\Sigma_{3}$ so that the positive
twist-and-glue maps associated to edges of $G$ are done before $e$. Applying
the previous paragraph to $G$, we produce a $pqr$-chain on $G$. Since $e$
connects two vertices of $G$, the map it induces is applied to a pair of loops
already contained in the $pqr$-chain. When these loops are both $x$-labeled,
the summand vanishes. Otherwise, one loop is $x$-labeled and the other
$1$-labeled, and the result is the enhanced state with all $x$-labeled
$\Gamma_{0}$. There are exatly two summands in the $pqr$-chain where these
loops are not both $x$-labeled, so the result of applying $e$ is twice the all
$x$-labeled $\Gamma_{0}$. The sign is the product of the signs of these
summands.

When $b^{1}(\Gamma_{0})\geqslant 2$, apply the previous paragraph and record
an additional edge that is not in $G$. Applying the positive twist-and-glue
map associated to this edge will kill the enhanced state we have just
produced. ∎

######  Remark 4.7.

In the case that $\Gamma_{0}$ is not connected, the theorem may be
individually applied to each component; the positive twist-and-glue moves
associated to different components act on labels for distinct Seifert circles
(if they shared a Seifert circle, the two components would share a vertex).

###  4.4. Nontriviality of Relative Khovanov-Jacobsson Classes

In the following sections, we will use the nontriviality of relative Khovanov-
Jacobsson classes to obstruct certain properties of surfaces bounding a link.
It is then important to establish conditions for determining when cycles in
the Khovanov chain complex represent nontrivial homology classes. Conditions
specific to enhanced state cycles are discussed in [Ell09] and [Ell10].
Unfortunately, these conditions are not exhaustive. Moreover, relative
Khovanov-Jacobsson cycles are often linear combinations of enhanced states,
with each term not necessarily constituting a cycle. Further development of
these conditions would be promising for future applications of relative
Khovanov-Jacobsson classes.

One method that has proven useful for determine the nontriviality of some
cycles is the following: if it is difficult to determine whether the relative
Khovanov-Jacobsson cycle of a link cobordism $\Sigma\colon\varnothing\to L$ is
nontrivial, we can extend $\Sigma$ by a second link cobordism
$\Sigma^{\prime}\colon L\to L^{\prime}$ and hope that the nontriviality of
$\textnormal{KJ}_{\Sigma^{\prime}\circ\Sigma}$ is simpler to determine in
$\textnormal{Kh}(L^{\prime})$. Because
$\textnormal{KJ}_{\Sigma^{\prime}\circ\Sigma}=\textnormal{Kh}(\Sigma^{\prime})(\textnormal{KJ}_{\Sigma})$
and $\textnormal{Kh}(\Sigma^{\prime})$ is a homomorphism, a nontrivial
$\textnormal{KJ}_{\Sigma^{\prime}\circ\Sigma}$ necessitates a nontrivial
$\textnormal{KJ}_{\Sigma}$.

One link cobordism $L\to L^{\prime}$ we will use is called a trim cobordism.
This link cobordism removes a single crossing from the diagram for $L$,
thereby reducing the complexity of the Khovanov homology group in which
nontriviality is being determined. A crossing is removed by attaching a
$1$-handle and undoing the resulting kink with a Reidemeister I move. This is
illustrated for a right-handed crossing in Figure 6 and for a left-handed
crossing in Figure 7. The $1$-handle can be attached on either the right or
left of the crossing; the computations are symmetric, so we have only shown
the left-sided attachment. The map induced by a trim cobordism is called a
trim map.

######  Remark 4.8.

The reader may be alarmed that the $1$-handle used to perform a trim cobordism
may produce a nonorientable link cobordism. Because trim maps are only used
for determining nontriviality, the orientability of the link cobordism $L\to
L^{\prime}$ is unimportant: nonoriented Morse saddles still induce chain maps.
A choice of orientation for the link $L^{\prime}$ simply determines the
grading of the chain map, and therefore, will not effect the nontriviality of
the cycle.

######  Remark 4.9.

Trim maps regularly produce cycles representing the trivial homology class.
Only clever and careful use of trim maps provide useful insights!

##  5\. Obstructing Sliceness of Knots

###  5.1. Techniques for Obstructing Sliceness

In certain cases, relative Khovanov-Jacobsson classes can be used to determine
the sliceness of a knot. In particular, by implementing the characterization
of absolute Khovanov-Jacobsson classes in Theorem 3.1 , we obtain the
following result.

######  Theorem 5.1.

For a slice knot $K$, the relative Khovanov-Jacobsson class of a link
cobordism $\Sigma\colon\varnothing\to K$ is nontrivial if $g(\Sigma)\leqslant
1$. In particular, slices of a slice knot have nontrivial relative Khovanov-
Jacobsson classes.

Proof. First, suppose $g(\Sigma)=0$. Produce an oriented, genus $1$, closed
surface $T=-(\Sigma\\#T^{2})\circ\Sigma$. By the characterization of absolute
Khovanov-Jacobsson classes, $\textnormal{Kh}(T)(1)=2$. Therefore,

$2=\textnormal{Kh}(T)(1)=\big{(}\textnormal{Kh}(-(\Sigma\\#T^{2}))\circ\textnormal{Kh}(\Sigma)\big{)}(1)=\textnormal{Kh}(-\Sigma\\#T^{2})(\textnormal{KJ}_{\Sigma}).$

A trivial $\textnormal{KJ}_{\Sigma}$ kills the right-most term, producing a
contradiction. The proof with $g(\Sigma)=1$ is done similarly; compose
$\Sigma$ with a slice of $K$ to yield an identical calculation. ∎

Figure 6. The left trim map on a right-handed crossing. Figure 7. The left
trim map on a left-handed crossing.

###  5.2. Obstructing Sliceness of Pretzel Knots

Here, we use relative Khovanov-Jacobsson classes to obstruct the sliceness of
certain three stranded pretzel knots. The sliceness of all odd three-stranded
pretzel knots was determined in [GJ11], and for the family we consider, the
signature provides a more direct obstruction to sliceness. Nevertheless, this
result indicates the potential for relative Khovanov-Jacobsson classes to
obstruct sliceness more generally.

######  Theorem 5.2.

For $p,q,r$ odd and of the same sign, $P(p,q,r)$ is not slice.

Proof. A knot is slice if and only if its mirror is slice; since the mirror of
$P(p,q,r)$ is $P(-p,-q,-r)$, it suffices to prove the case where $p$, $q$, and
$r$ are all negative. We will show that the Seifert surface $\Sigma_{D}$
associated to the standard diagram $D$ in Figure 8 has trivial relative
Khovanov-Jacobsson, whereby Theorem 5.1 implies $P(p,q,r)$ is not slice. To
show that $\Sigma_{D}$ has trivial relative Khovanov-

Figure 8. An oriented $P(-5,-3,-5)$; the orientation-induced smoothing
$\sigma_{D}$; the subgraph $\Gamma_{0}$ of the state graph
$\Gamma_{\sigma_{D}}$, having first Betti number $2$.

Jacobsson class, we apply Theorem 4.6 , which characterizes the relative
Khovanov-Jacobsson cycles for Seifert surfaces according to the Betti number
of the subgraph $\Gamma_{0}$ of their state graph $\Gamma_{\sigma_{D}}$.
Either orientation induces the $0$-state of $D$ whose subgraph
$\Gamma_{0}=\Gamma_{\sigma_{D}}$ has first Betti number $2$. Thus, the
relative Khovanov-Jacobsson cycle for $\Sigma_{D}$ is trivial. We conclude
$P(p,q,r)$ is not slice. ∎

######  Remark 5.3.

In an attempt to reprove the result of [GJ11], we could consider the case
where one parameter $q$ is positive. In this case,
$b^{1}(\Gamma_{\sigma_{D}})=1$ so $\textnormal{KJ}_{\Sigma}$ is twice the
enhanced state having $x$-labels on the vertices of $\Gamma_{0}$ and
$1$-labels otherwise. It is unclear if this enhanced state represents a
nontrivial Khovanov homology class. On the other hand,
$\textnormal{KJ}_{\Sigma}$ is nontrivial if $q=-p$ or $q=-r$ because
$P(p,-p,r)$ and $P(p,-r,r)$ are both slice. We suspect that this is the case
for all $P(p,q,r)$ with positive $q$ because the graph structure of
$\Gamma_{0}$ is unchanged by the value of the parameters.

######  Remark 5.4.

Odd parameters for a pretzel link always produce a knot. Using one even
parameter will also be a knot. Unfortunately, in this case, the Seifert
surface rarely has Euler characteristic $0$ or $-1$, so the above argument is
not applicable to this situation.

##  6\. Obstructing Boundary-Preserving Isotopy Classes

###  6.1. Techniques for Obstructing Boundary-Preserving Isotopy Classes

Recall that relative Khovanov-Jacobsson classes are invariants of the
boundary-preserving isotopy class of a link cobordism: two link cobordisms
$\Sigma_{0,1}\colon\varnothing\to L$ whose relative Khovanov-Jacobsson classes
are distinct, $\textnormal{KJ}_{\Sigma_{0}}\neq\textnormal{KJ}_{\Sigma_{1}}$,
are not isotopic relative to their boundary. To obstruct this isotopy, we show

$\mathcal{C}(\Sigma_{0},\Sigma_{1}):=\pm\mathcal{C}\textnormal{KJ}_{\Sigma_{0}}\pm\mathcal{C}\textnormal{KJ}_{\Sigma_{1}}$

represent nontrivial Khonvanov homology classes.

###  6.2. Local Knottedness

Given a link cobordism $\Sigma\colon\varnothing\to L$ and a knotted $2$-sphere
$S$, we say $\Sigma\\#S$ is locally knotted. This operation generally changes
the boundary-preserving isotopy class of $\Sigma$, and as there is a plethora
of knotted $2$-spheres with which we can perform this operation, the study of
boundary-preserving isotopy classes is more rewarding when this operation is
omitted. For a knot $K$, let $\overline{\Sigma(K)}$ denote the set of
equivalence classes of surfaces bounding $K$ under the equivalence relation
generated by boundary-preserving isotopy and connect summing with a knotted
$2$-sphere. For a link cobordism $\Sigma\colon\varnothing\to K$ we let
$\Sigma(K)$ denote its equivalence class in $\overline{\Sigma(K)}$.

Recent work has drawn attention toward understanding $\overline{\Sigma(K)}$
for classes of slice knots; see [JZ20], [CP20], and [MP20]. The latter of
these posed the question: can we determine the value of
$|\overline{\Sigma(K)}|$ for some nontrivial knot $K$, or at least whether
$|\overline{\Sigma(K)}|<\infty$. The following Theorem and Corollary indicate
that relative Khovanov-Jacobsson classes may provide some aid in calculating
lower bounds for this value.

######  Theorem 6.1.

Relative Khovanov-Jacobsson classes do not detect local knottedness.

Proof. For a link cobordism $\Sigma\colon\varnothing\to L$ and $2$-knot $S$,
we wish to show $\textnormal{KJ}_{\Sigma}=\textnormal{KJ}_{\Sigma\\#S}$. Let
$E$ and $F$ be the disks removed from $\Sigma$ and $S$, respectively, to form
$\Sigma\\#S$. This allows us to write $\Sigma\\#S$ as a composition of
cobordisms $(\Sigma\setminus E)\circ(S\setminus F)$. In forming the connect
sum, the disk $E$ is replaced by $S\setminus F$. As both of these are genus
$0$ surfaces bounding the unknot, Theorem 4.2 implies they induce the same map
on Khovanov homology, whereby we have

$\textnormal{KJ}_{\Sigma}=\textnormal{Kh}_{(\Sigma\setminus E)\circ
E}(1)=\big{(}\textnormal{Kh}_{\Sigma\setminus
E}\circ\textnormal{Kh}_{E}\big{)}(1)=\big{(}\textnormal{Kh}_{\Sigma\setminus
E}\circ\textnormal{Kh}_{S\setminus
F}\big{)}(1)=\textnormal{Kh}_{(\Sigma\setminus E)\circ(S\setminus
F)}(1)=\textnormal{KJ}_{\Sigma\\#S}$

Therefore, local knottedness is not detected by relative Khovanov-Jacobsson
classes. ∎

Although the statement of this theorem may sound like a shortcoming of
relative Khovanov-Jacobsson classes, it actually gives them the power to
distinguish classes in $\overline{\Sigma(K)}$. In particular, whenever a pair
of surfaces bounding a knot $K$ have distinct relative Khovanov-Jacobsson
classes, it must follow that the surfaces represent distinct classes in
$\overline{\Sigma(K)}$, lest they contradict the above theorem. More
succinctly, we have the following result.

######  Corollary 6.2.

Relative Khovanov-Jacobsson classes are invariants of classes in
$\overline{\Sigma(K)}$ for any $K$.

###  6.3. Obstructing Equivalence of Slice Disks for $\boldsymbol{9_{46}}$

There are two slices $\Sigma_{\ell}$ and $\Sigma_{r}$ for the knot $9_{46}$,
obtained by first performing either the left (pink) or right (orange) band
move, indicated

on the diagram to the right, and then undoing the resulting two-component
unlink. It is known that these slices are not isotopic relative to their
boundary. In what follows, we verify this fact by showing these slices have
distinct relative Khovanov-Jacobsson classes. We go on to show that
$\\#_{k}(9_{46})$ has $2^{k}$ distinct slices, implying
$|\overline{\Sigma\big{(}\\#_{k}(9_{46})\big{)}}|\geqslant 2^{k}$. This fact
was also established in [MP20] by examining kernels of inclusion induced maps
on Alexander modules.

######  Theorem 6.3.

The pretzel knot $P(3,-3,3)=9_{46}$ has at least two distinct slices.

Proof. A movie of the surface $\Sigma_{r}$ is given in Figure 10 (found at the
end of the paper), along with the relative Khovanov-Jacobsson cycle it
produces. A symmetric calculation gives the relative Khovanov-Jacobsson cycle
for $\Sigma_{\ell}$. Both cycles are illustrated in Figure 9 . We claim that
the up-to-sign differences of these cycles
$\mathcal{C}(\Sigma_{\ell},\Sigma_{r})$ represent nontrivial homology classes,
and therefore the boundary-preserving isotopy classes of $\Sigma_{\ell}$ and
$\Sigma_{r}$ are distinguished by their relative Khovanov-Jacobsson classes.

Figure 9. The relative Khovanov-Jacobsson cycles for $\Sigma_{\ell}$ (top) and
$\Sigma_{r}$ (bottom).

######  Remark 6.4.

Although the movie descriptions for these surfaces produce cycles belonging to
Khovanov chain complexes associated to the same diagram of $9_{46}$, these
chain complexes differ in their enumeration of the diagram’s crossings. In
retrospect, this will not be important, however, for the general well-being of
the reader, we note that these complexes are related by a chain homotopy that
only affects the signs of the enhanced states in
$\mathcal{C}\textnormal{KJ}_{\Sigma_{\ell}}$ and
$\mathcal{C}\textnormal{KJ}_{\Sigma_{r}}$. There are no common smoothings
between these cycles, so these sign changes will not cause any unwanted
cancellation.

To see $\mathcal{C}(\Sigma_{\ell},\Sigma_{r})$ represents a nontrivial element
in homology, apply three positive trim maps to the crossings in the left
column of the diagram, killing any enhanced state having a $1$-smoothing in
the left column. There is only one enhanced state whose left column contains
only $0$-smoothings; we have highlighted this state (in purple) in
$\mathcal{C}\textnormal{KJ}_{\Sigma_{r}}$. As a result, the three trim maps
produce a single enhanced state. This enhanced state is the all $1$-label of
the $1$-state of the trimmed diagram;

Figure 10. A movie description of the surface $\Sigma_{r}$ and the calculation
of its relative Khovanov-Jacobsson cycle. Highlighted is the element which
maintains a left column that is all $0$-smoothed.

an analysis on the differentials entering this (extreme) homological grading
shows this enhanced state cannot be a boundary. Details for this analysis can
be found in both [Ell09, Example 2.2] and [Swa10, Proposition 10].

Unpackaging the proof: this enhanced state is the image of
$\mathcal{C}(\Sigma_{\ell},\Sigma_{r})$ under a sequence of trim maps and
represents a nontrivial class in the Khovanov homology associated to the
trimmed diagram. Thus, the cycles $\mathcal{C}(\Sigma_{\ell},\Sigma_{r})$ must
also represent nontrivial Khovanov homology classes, forcing their relative
Khovanov-Jacobsson classes to be distinct. We may then conclude that the
slices $\Sigma_{\ell}$ and $\Sigma_{r}$ are not isotopic relative to their
boundary. ∎

######  Corollary 6.5.

For $n$ odd and $|n|>1$, the pretzel knot $P(n,-n,n)$ has at least two
distinct slices.

Proof. Use a similar diagram with $n$-crossings in each column and alter the
above movie descriptions to use $n$-many Reidemeister I and II moves. The
relative Khovanov-Jacobsson chain cycles are more complex, however, throughout
their calculation, we can follow a unique enhanced state having a left column
that is all $0$-smoothed (just as we did in purple throughout Figure 10 ).
Applying $n$-many positive trim moves to the left column of the diagram will
kill all but this unique enhanced state, and, as before, this enhanced state
will be the all $1$-label of the $1$-state of the trimmed diagram, which
represents a nontrivial Khovanov homology class. ∎

######  Corollary 6.6.

The knot $\\#_{k}(9_{46})$ has at least $2^{k}$-many distinct slices.

Figure 11. The windmill diagram (left) for $\\#_{5}(9_{46})$ and an enhanced
state windmill (right) from the relative Khovanov-Jacobsson cycle associated
to one of its slices.

Proof. Use a windmill diagram such as the one in Figure 11 , with $k$-many
inward-facing copies of $9_{46}$ as the sails of the windmill and $k$-many
windshafts which connect the sails in the center of the diagram. Movies for
the $2^{k}$-many slices are obtained by choosing either of the movies from
Theorem 6.3 for each sail and using $k$-many Morse saddles to merge their
final frames along the windshafts.

By connecting the diagrams in this way, the relative Khovanov-Jacobsson cycle
associated to one of these movies consists of $8^{k}$-many enhanced state
windmills whose sails are one of the $8$-many enhanced states from
$\mathcal{C}\textnormal{KJ}_{\Sigma_{\ell}}$ or
$\mathcal{C}\textnormal{KJ}_{\Sigma_{r}}$, depending on which movie was chosen
for the corresponding sail in the diagram. The enhanced state windmills have
this form because the final $k$-many Morse saddles forming the windshafts will
connect the enhanced states from $\mathcal{C}\textnormal{KJ}_{\Sigma_{\ell}}$
and $\mathcal{C}\textnormal{KJ}_{\Sigma_{r}}$ along the “top loop” in the
smoothing; because all of these loops are $1$-labeled, the Morse saddles will
preserve this labeling. In the aforementioned figure, we have illustrated an
enhanced state windmill for the slice of $\\#_{k}(9_{46})$ having movies of
$\Sigma_{r}$ on all but one sail.

A more general way to describe these slices is as the boundary-connected sum
of $k$-many copies of $\Sigma_{\ell}$ or $\Sigma_{r}$. The utility of the
(longwinded) process above is a specified movie description.

We now prove the result. Choose a pair of slices $\Sigma_{0,1}$ from the
$2^{k}$-many slices described above, and produce their relative Khovanov-
Jacobsson cycles per the given movie descriptions. As in the proof of Theorem
6.3 , we will trim the diagrams to simplify the cycles, although in this
case, we must trim each sail. Trim the left column of a sail when the movie of
$\Sigma_{0}$ shows $\Sigma_{r}$ and trim the right column when it shows
$\Sigma_{\ell}$. Because $\Sigma_{0}$ and $\Sigma_{1}$ differ in their choice
of movies for the sails, one of the trims will kill
$\mathcal{C}\textnormal{KJ}_{\Sigma_{1}}$. The only remaining element of
$\mathcal{C}\textnormal{KJ}_{\Sigma_{0}}$ will be the all $1$-label of the
$1$-state of the trimmed diagram of $\\#_{k}(9_{46})$. For example, we have
shown the trim of the enhanced state windmill from Figure 11 here. As before,
such an element represents a nontrivial homology class in the Khovanov
homology of this trimmed diagram. The result follows just as it did in the
previous two results. ∎

######  Corollary 6.7.

The knot $\\#_{k}\big{(}P(n,-n,n)\big{)}$ has at least $2^{k}$-many distinct
slices.

Proof. Similar to Corollary 6.5 , the number of crossing used in the strands
for each sail of the windmill does not affect the argument in Corollary 6.6 .
∎

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