TOPICS IN LINK HOMOLOGY
By
Thomas Constantin Jaeger

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Mathematics
2011

ABSTRACT
TOPICS IN LINK HOMOLOGY
By
Thomas Constantin Jaeger
We prove two results about mutation invariance of link homology theories: We show that
Khovanov’s universal sl(2) homology is invariant under mutation and that the reduced sl(n)
homology defined by Khovanov and Rozansky is invariant under component-preserving positive mutation when n is odd. We also give a relationship between the Khovanov homology
of a closed positive 3-braid and the Khovanov homology of the braid after adding a number
of full twists.

ACKNOWLEDGMENTS

I would like to thank my adviser, Effie Kalfagianni, for her expertise, support and encouragement. I am grateful to Matt Hedden and Mikhail Khovanov for their valuable comments
on earlier versions of this work and to Teena Gerhardt for additional comments. I am indebted Ben Cooper for pointing me toward [7] when I needed to plug a hole in one of the
proofs. I would also like to thank my fellow graduate students Jeff Chapin, Andrew Cooper,
Chris Cornwell, Chris Hays, Daniel Smith, Nathan Sunukjian and Carlos Vega. Last but not
least, I am thankful to Cheryl Balm and Ittles McGee for their moral support and clerical
assistance.

iii

TABLE OF CONTENTS

List of Figures

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1 Introduction

v
1

2 Definitions
2.1 Mutation . . . . . . . . . . . . . . .
2.2 Enriched Categories . . . . . . . . .
2.3 Universal sl(2) Khovanov Homology
2.4 Khovanov-Rozansky homology . . .

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3 Link Homology and Mutation
3.1 Lemma about invariance of mapping cones . . . . . . . . . . . . . . . . . .
3.2 Mutation invariance of sl(2) homology . . . . . . . . . . . . . . . . . . . .
3.3 Mutation invariance of sl(n) homology . . . . . . . . . . . . . . . . . . . .
3.3.1 Topological considerations . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Behavior of the Khovanov-Rozansky chain complex under reflection
3.3.3 Khovanov-Rozansky Homology of 2-tangles . . . . . . . . . . . . . .
3.3.4 Mutation invariance of the inner tangle . . . . . . . . . . . . . . . .
3.3.5 Proof of invariance . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Khovanov Homology of 3-braids
4.1 Khovanov Homology . . . . . . . . . .
4.2 Simplifying chain complexes . . . . . .
4.3 Three-Braids . . . . . . . . . . . . . .
4.4 Khovanov homology after adding twists

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LIST OF FIGURES

2.1

Placement and labeling of the endpoints of the inner tangle . . . . . . . . . .

8

2.2

A 2-tangle with orientation-reversing symmetry and the Kinoshita-Terasaka Conway mutant pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

Positive crossing, negative crossing, oriented smoothing and singular crossing.
The dotted line connecting the two arcs of the oriented smoothing illustrates
that we consider both arcs to be on the same component of the smoothing. .

19

3.1

Proof of Lemma 3.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

3.2

The two possible orientations of the endpoints of inner tangle

. . . . . . . .

38

3.3

Rz mutation on a tangle of type (b) is equivalent to Ry mutation on a tangle
of type (a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

3.4

A closure of the tangle and its Seifert picture . . . . . . . . . . . . . . . . . .

39

3.5

Transforming the diagram into braid form . . . . . . . . . . . . . . . . . . .

39

3.6

Singular braid diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

4.1

2
Transforming ∆ sX into sX . All other cases, negative twist, first B-smoothing
on the bottom are analogous. . . . . . . . . . . . . . . . . . . . . . . . . . .

64

4.2

Reidemeister II moves induce special deformation retracts . . . . . . . . . . .

65

4.3

The Khovanov complex near a corner of the Khovanov cube . . . . . . . . .

69

4.4

Decomposing the Khovanov complex of a closed braids into a direct sum . .

70

4.5

(Partial) tree of resolutions of ∆

2.3

2

. . . . . . . . . . . . . . . . . . . . . . . .

v

71

Chapter 1
Introduction
This thesis is focused on properties of link homology theories. Link homology theories can
be viewed as refinements of polynomial invariants associated to links.
In the 1980s, the discovery of the Jones polynomial [15] sparked an interest in polynomial
invariants of knots, leading to various generalizations of the Jones and the Alexander polynomials such as the HOMFLY-PT polynomial [12] and Reshetikhin and Turaev’s invariant
for links colored by Lie algebra representations [40].
Beginning in the late 1990s, various corresponding link homology theories were introduced, starting with Khovanov’s categorification of the Jones polynomial [24] and knot Floer
homology, a link homology theory discovered by Oszv´th and Szab´ [18] and independently
a
o
by Rasmussen [39], whose graded Euler characteristic recovers the Alexander polynomial.
The idea of categorifying link polynomials has been especially fruitful in constructing such
link homology theories.
The goal of categorifying a mathematical object is to lift the construction one (categorical) dimension up, so that the original object can be recovered as a “shadow” of the
1

categorified object, hopefully explaining connections that seemed mysterious in the decategorified world. As an example, consider the natural numbers. Natural numbers are ubiquitous because we use them to count things; mathematically speaking, they are cardinalities
of finite sets. But by identifying all sets of the same cardinality, we lose the ability to talk
about relationships between the objects we are counting. A reasonable categorification of
the natural numbers is thus the category of finite sets with equality weakened to bijection
and addition and multiplication corresponding to disjoint union and direct product. A different categorification, which requires us to make fewer choices on the level of objects, is the
category of finite-dimensional vector spaces over a fixed field, where the number assigned
to a vector space is simply its dimension. An advantage of vector spaces over sets is that
this categorification naturally extends to a categorification of the integers by passing from
vector spaces to chain complexes, where the underlying integer can be recovered by taking
the Euler characteristic of the complex. Natural operations on the integers readily carry over
into the categorified setting: Addition corresponds to direct sums, multiplication to tensor
products and negation corresponds to an odd shift in homological degree.

Polynomials are collections of integers living in various degrees, hence what we are looking
for in a categorification of a link polynomial is a chain complex of graded vector spaces (or
more generally graded modules over a ring) for each diagram of a link. But there should be
additional structure: At the very least, every isomorphism of link diagrams, that is every
sequence of Reidemeister moves, should induce an isomorphism (in a suitable sense) of graded
chain complexes. In fact, we are asking for more: Every cobordism of links, represented by a
sequence of Reidemeister moves and handle attachments, should correspond to a chain map
which, up to homotopy, is independent of the presentation of the cobordism. In practice,
2

this is often only true projectively, that is up to a sign.
Further structure is revealed by climbing down one step on the dimensional ladder. In
the case of link polynomials, we associate a module to a circle with an even number of
marked points (thought of as a diagram for a two-sphere with marked points) together with
multilinear maps corresponding to all the ways of gluing disks with marked points on the
boundary together to form a new such disk. This is called a planar algebra. To every tangle
diagram, we associate an element of the module associated to its boundary; in particular,
the module assigned to a circle without marked points is free of rank 1 and the element of
this module associated to a link recovers the link polynomial. This allows us to compute
the invariant of the link by cutting and pasting: once we know the value of the invariant
on the basic building blocks, i.e. the crossings, we can use the planar algebra structure to
compute the invariant for arbitrary links and tangles. We may also need to take orientations
into account, which is done by turning marked points into arrows that can point inward or
outward.
Similar structure can often be found in the categorified setting. To a circle with marked
points or arrows, we associate a triangulated category, usually the homotopy category of
chain complexes over some additive category, together with tensor products corresponding to
gluing of disks. To every tangle diagram we associate an object of the category associated to
its boundary and to every cobordism of tangles we associate a morphism in the corresponding
category. This allows the calculation of the invariant by the same cut-and-paste approach
described above.
A major part of this thesis is concerned with the effect mutation has on link homologies.
Mutation is a topological operation that transforms a link into another link by cutting out
3

a ball that meets the link in four points, applying an involution of the ball that preserves
the set of intersection points of the link with the boundary of the ball and then gluing the
ball back in. This operation has no effect on many link polynomials, a fact that is usually
remarkably easy to see. The situation is more complicated in the case of link homology theories, which often detect mutation of links even when the underlying link polynomial does
not. We study two of those link homology theories. The first is universal Khovanov homology, a generalization of Khovanov’s original link homology theory categorifying the Jones
polynomial. An advantage of working in this more general setting is that it contains enough
information to recover various spectral sequences that can be used to define Rasmussen’s
s-invariant and other similar invariants that are useful to study the slice genus of knots. We
succeed in showing invariance of universal Khovanov homology when working in characteristic 2. As an immediate corollary, we get invariance of a certain analog of the s-invariant,
but we do not know whether it coincides with Rasmussen’s original invariant. The second
theory that we study is sl(n) homology, a theory defined by Khovanov and Rozansky as a
categorification of the sl(n) polynomial. We show that, for odd n, the reduced version of
this theory is invariant under mutation when imposing certain restrictions on the kinds of
mutations that we allow.

The last chapter of the thesis investigates the Khovanov homology of links represented
by 3-braids. We describe an explicit relationship between the Khovanov homology of the
closure of a positive 3-braid and closure of the same 3-braid after adding a number of full
twists. Every closed 3-braid can be written in this form and all but three 3-braids can be
represented (up to conjugation) as the product of an alternating braid and a number of
twists, whose Khovanov homology is determined by the Jones polynomial and the signature.
4

We point out that although we will happily call the theories we are considering “homology
theories,” we will rarely actually take homology and rather work with the underlying chain
complexes (up to homotopy) instead. The reason for this is two-fold. One, the tangle
invariants are chain complexes over an additive category that is in general not abelian.
Two, experience has taught us that the proper setting to study homological algebra in is
the derived category and homology may forget some of the information that is present in
the derived category. The homotopy categories that our tangle invariants live in are in
some sense analogs of the derived category. This can be made precise in the case of links,
where we are working in the homotopy category of chain complexes of free modules over
some graded polynomial ring, which is (at least in the bounded above case) equivalent to the
derived category of graded modules over the ring (see for example [46], Theorem 10.4.8). The
difference between “free” and projective objects vanishes here because of the Quillen-Suslin
Theorem.
Finally, we remark that it has become common in the literature to refer to the theories as
“homology theories” despite the fact that they are, strictly speaking, cohomology theories.
We adopt this convention and do not worry about the distinction unless we are dealing with
(co)homology of topological spaces. This is justified because the notions are identical up to
re-indexing. All chain complexes we will consider are cohomological, that is the differential
increases (co)homological degree.
The organization of this dissertation is as follows: In chapter 2, we review the definitions
of the link homology theories that are the topic of this dissertation, specifically (universal)
Khovanov homology and sl(n) homology. We then investigate the behavior of these theories
under mutation in chapter 3, showing that universal Khovanov homology of knots is invariant
5

in characteristic 2 and that for odd n, reduced Khovanov-Rozansky homology of knots is
invariant under a particular type of mutation. In chapter 4, we analyze the effect that adding
a twist has on the Khovanov homology of a positive 3-braid.

6

Chapter 2
Definitions
In this chapter, we review the definitions we will use throughout the thesis. Whenever
definitions differ from the definitions in the literature, we show that both are equivalent.

2.1

Mutation

Conway mutation is the process of decomposing a link L as the union of two 2-tangles
L = T ∪ T and then regluing in a certain way. Diagrammatically, we may assume that one
of the tangles (the ‘inner’ tangle T ) lies inside a unit circle with endpoints equally spaced
as in Figure 2.1. Mutation consists of one of the following transformations R of the inside
tangle, followed by regluing: reflection along the x-axis (Rx ), reflection along the y-axis (Ry )
or rotation about the origin by 180 degrees (Rz ). In other words, the mutant is given by
L = R(T ) ∪ T . When taking orientations into account, we can distinguish two types of
mutation (see for example Kirk and Livingston [17]).
Definition 2.1.1. Mutation of an oriented link is called positive if orientations match when
regluing, i.e. if L = R(T ) ∪ T as an oriented link and it is called negative if the orientation
7

c

y

d
x

a

b

Figure 2.1: Placement and labeling of the endpoints of the inner tangle

c

d

a

b
(a) T

n
(b) 1142

n
(c) 1134

Figure 2.2: A 2-tangle with orientation-reversing symmetry and the Kinoshita-Terasaka Conway mutant pair
of the inner tangle needs to be reversed before regluing, i.e. if L = −R(T )∪T as an oriented
link, where −R(T ) denotes R(T ) with orientations reversed.
n
n
As an example, consider the two knots in Figure 2.2. 1134 is a positive mutant of 1142
since rotation about the x-axis preserves the orientations of the ends of T . It is also a
negative mutant, as can be seen by considering rotation about the z-axis.
There are 16 mutant pairs with 11 or fewer crossings, see [30]. It can be checked that
all of them can be realized by negative mutation. Among the 16 pairs, we found 5 that
a
a
a
a
can be realized on the tangle T depicted in Figure 2.2(a): (1157 , 11231 ), (11251 , 11253 ),
n
n
n
n
n
n
(1134 , 1142 ), (1176 , 1178 ) and (11151 , 11152 ). Ry (T ) is isotopic to T but with orientations
reversed, therefore these 5 mutant pairs can be realized by both positive and negative mutation. In particular, our proof applies to the famous Kinoshita-Terasaka - Conway pair,
illustrated in Figure 2.2(b) and (c).

8

Definition 2.1.2. Mutation of a link is called component-preserving if a and R(a) lie on the
same component of the original link (or equivalently, on the same component of the mutant).
Note that knot mutation is always component-preserving. If positive mutation is component-preserving, then a and R(a) are either both incoming or both outgoing edges, hence
all 4 endpoints lie on the same component of the link. We refer to this component as the
component of the mutation.

2.2

Enriched Categories

We will adopt the language of enriched categories ([9]) for our description of universal sl(2)
homology. An enriched category A over a symmetric monoidal category (V, ⊗, I), or Vcategory, is a collection of objects together with an assignment of an object A(A, B) of V to
every pair of objects A and B, an identity I → A(A, A) for every object A and composition
A(B, C) ⊗ A(A, B) → A(A, C) for each triple of objects A, B and C. These are required
to satisfy certain natural axioms, see [9] for details. Most of the constructions below can
be carried out in much greater generality, but for simplicity, we restrict ourselves to two
examples of categories V: The category R-mod of R-modules over a commutative ring R
and the category R-gmod of graded R-modules, where R is a graded commutative ring.
Morphisms in R-gmod are required to be of degree 0. We introduce a grading shift functor
{·} : R-gmod → R-gmod such that if x ∈ A, then the corresponding element y ∈ A{k}
satisfies deg y = deg x + k.
An ideal I in an R-mod-category A is a collection of submodules I(A, B) ≤ A(A, B)
such that ρψφ ∈ I(A, D) for φ ∈ A(A, B), ψ ∈ I(B, C) and ρ ∈ A(C, D). It is easy to
see that given an ideal I in an R-mod-category A, there is a quotient category A/I whose
9

objects are the objects of A and morphism spaces are (A/I)(A, B) = A(A, B)/I(A, B).
Let G be a directed multigraph. The free R-mod-category A on G has objects the vertices
of G and morphism spaces A(A, B) the free R-module generated by all paths from A to B
in G. Composition is given by concatenation of paths using the identification R ⊗R R ∼ R.
=
If R is a graded ring and we assign a degree to each edge of G, then the free R-mod category
on G has the structure of an R-gmod-category, which we will refer to as the free R-gmodcategory on G. More precisely, a path φn · · · φ1 from A to B corresponds to a summand
R{deg φ1 +· · ·+deg φn } in A(A, B). An ideal I of A is called homogeneous if for each A and
B, the homogeneous parts of elements of I(A, B) are also in I(A, B). It is straightforward to
see that the ideal generated by homogeneous elements is homogeneous and that the quotient
category by a homogeneous ideal is again an R-gmod-category.
If A is an R-gmod-category, we define A to be the R-gmod category whose objects are
formal shifts A{k} of objects A of A, with morphism spaces given by A(A{k}, B{l}) =
A(A, B){l − k}.
Given a V-category A, where V is either R-mod or R-gmod, we define an underlying
category A0 , whose objects coincide with the objects of A and whose morphisms A → B are
(degree-0) elements of the (graded) R-module A(A, B). In both cases A0 is a pre-additive
category.
For any pre-additive category C we define its additive closure Mat(C) to be the category
whose objects are (possibly empty) formal direct sums of objects of C and whose morphisms
are matrices of morphisms of C with the appropriate source and target objects. Composition
is given by matrix multiplication. Clearly, Mat(C) is additive and C embeds into Mat(C).
As a simple example, consider the graph G with one vertex and no edges. Let A be
10

the free R-gmod-category on G. A is the graded free R-module R of rank 1, regarded as
an R-gmod-category with one object. Its underlying category A0 is the full subcategory of
R-gmod on the object R. A0 is the category of free graded R-modules of rank 1, whereas
Mat(A0 ) is the category of finitely generated free graded R-modules, which we denote by
R-fgmod.

2.3

Universal sl(2) Khovanov Homology

We now define universal sl(2) homology. It is also known as equivariant sl(2) homology since
its underlying Frobenius system can be obtained by considering the U (2)-equivariant cohomology ring of the 2-sphere ([25]). We begin by describing the Frobenius system underlying
the construction the theory (a Frobenius system is a Frobenius extension together with a
choice of counit and comultiplication). This corresponds to the system F5 in [25]. Let R
be the ring Z[h, t] and A be the free R-module generated by

and

. We define a unit

ι : R → A, a counit : A → R and a multiplication map m : A ⊗R A → A by

ι(1) =
( )=0
( )=1
m(
m(

)=

) = m(

)=

m(

)=h
11

+t

Graphically, we may represent h, t, ι,

and m by

,

,

,

and

, respectively. The

composition ◦ m defines a symmetric bilinear form on A, giving an identification of A with
∗
its dual module A . We use the notation
and

for the image of

for the image of

under this identification

. Note that this is consistent with our previous notation for .

∗
∗
∗
If V and W are finitely generated free modules, then the map W ⊗ V → (V ⊗ W ) ,
∗
∗
∗
f ⊗ g → (v ⊗ w → f (w) · g(v)) gives a canonical identification of W ⊗ V with (V ⊗ W ) .
A comultiplication map ∆ : A → A ⊗R A, graphically represented by

, is obtained from

∗
∗
∗
∗
m : A → (A ⊗R A) using the identifications A ∼ A coming from the bilinear form ◦ m
=
∗
∗
∗
and the canonical identification (A ⊗R A) ∼ A ⊗R A . One readily checks that
=

∆( ) =

+

∆( ) =

−h

+t

Remark 2.3.1. Our treatment differs from that of Khovanov [25] in that we used self∗
duality (with respect to the identification of A and A coming from the bilinear form) of
∗
the Frobenius system as an axiom. If one uses the identification A → A ,
→

→

−h

,

, then the Frobenius system is only “almost self-dual”, as described in [25] in example

2 at the end of the section entitled “A universal rank two Frobenius system.”
The ring R becomes a graded ring by setting deg h = 2 and deg t = 4. We will refer
to this grading as q-grading. By setting the q-degree of the generator
q-degree of the generator

to be −1 and the

to be 1, we turn A into a graded R-module. One readily checks

that deg ι = deg = −1 and deg m = deg ∆ = 1. Note that the grading always corresponds
to the negative Euler characteristic of the associated picture.
The Frobenius system above defines a symmetric monoidal functor from the symmet12

ric monoidal category Cob with objects closed 1-manifolds and morphisms 2-dimensional
cobordisms modulo isotopy (with the monoidal structure given by disjoint union) to the
category R-mod of R-modules.
Associated to a link diagram D with n crossings is an n-dimensional cube of resolutions,
whose vertices are the resolutions of the diagram and whose edges are split or merge cobordisms corresponding to changing the resolution of a crossing from the 0-smoothing to the
1-smoothing. Adjusting signs so that faces anti-commute and flattening the cube, we obb
tain a complex in K (R-mod), the homotopy category of bounded complexes of R-modules.
This universal Khovanov complex Kh(D) is an invariant of the knot in the sense that if two
b
diagrams D and D represent the same knot, then Kh(D) ∼ Kh(D ) in K (R-mod). This
=
theory can be endowed with q-gradings, as explained for tangles below. Khovanov’s original
theory can be recovered as Kh(D)/(h = 0, t = 0).
This theory can be generalized to tangles along the lines of Bar-Natan [1]. Let D be
the diagram of a tangle, which we assume to lie inside a topological disk B and to have
boundary E := ∂D ⊂ B. We now define an R-gmod-category Kob(E) whose objects are
compact one-manifolds properly embedded in B with boundary E and whose morphisms
spaces are R-modules freely generated by equivalence classes of dotted cobordisms between
them. We require a cobordism f : X → Y to be properly embedded in B × [0, 1] and its
boundary to be E × [0, 1] ∪ X × {0} ∪ Y × {1}. Furthermore, we define the degree of a
generator E to be χ(f ) − 1 |E|, where dots contribute 2 to the Euler characteristic as above.
2
The equivalence relation on cobordisms is generated by isotopy and the graphical relations
from above. Explicitly:

• The value of a sphere without dots is 0, and the value of a sphere with one dot is 1.
13

This reflects how the counit

acts on generators.

• (Two-dot relation) Locally,
(

)=h

+t

+

+t

. This corresponds to the equation

.

• (Neck-cutting relation) Locally,
( )=

= h

−h

=

+

−h

. This corresponds to the equation

.

Let Kob (E) be the skeleton of the full subcategory of Kob(E) on one-manifolds without closed loops (two objects in an enriched category A are considered isomorphic if the
corresponding objects in A0 are isomorphic). Kob (E) has one object for every isotopy
class of such one-manifolds.
Let mKob(E) = Mat(Kob(E)0 ) and mKob (E) = Mat(Kob (E)0 ). Each object of
mKob(E) is isomorphic to an object of mKob (E) via the delooping isomorphism below
(compare Bar-Natan [3]), hence the two categories are equivalent.

−h
ggg3 ∅{1} ‡‡‡‡‡‡‡‡
‡‡‡‡‡
ggggg
ggggg
+
‡‡‡‡‡
g3
‡‡‡‡‡
ggggg
g
‡‡+
ggggg
∅{−1}

Gluing gives rise to a tensor product. Specifically, let B1 and B2 be two disks which share
a boundary arc and E1 and E2 be a collection of points on the boundary of B1 and B2 such
that E1 ∩B2 = E2 ∩B1 = {x1 , . . . xn }. If X1 and X2 are objects of Kob(E1 ) and Kob(E2 ),
respectively, we define the tensor product X1 ⊗R[x ,...x ] X2 ∈ Kob((E1 ∪E2 )∩∂(B1 ∪B2 ))
1
n
to be X1 ∪X2 . Similarly, for morphisms fi : Xi → Yi (i = 1, 2), we define f1 ⊗R[x ,...x ] f2 :
1
n
X → Y , where X = X1 ⊗R[x ,...x ] X2 and Y = Y1 ⊗R[x ,...x ] Y2 , to be f1 ∪ f2 .
1
n
1
n
b
We can now define the Khovanov complex Kh(D) ∈ K (mKob(E)) for any tangle
14

diagram D inside a disk B with endpoints E by specifying the Khovanov complex of a
single crossing and then extending to arbitrary tangles using the tensor product above. The
complexes for a single positive and a single negative crossing are shown below. Underlined
chain objects represent 0 homological height.

Kh( ) :

{1} − − −
− −→

{2}

We use the two-dimensional pictures

{−2} − − −
− −→

Kh( ) :

and

{−1}

to represent the cobordisms in the dif-

b
ferentials of Kh( ) and Kh( ), respectively. Since the categories K (mKob(E)) and
b
K (mKob (E)) are equivalent with an explicit equivalence induced by the delooping funcb
tor, we will often regard Kh(D) simply as an object of K (mKob (E)), in particular we
b
consider Kh(D) of a link diagram D to be an object of K (R-fgmod).
Every edge of a tangle diagram, which we represent by a point p on the edge, induces
a chain morphism xp : Kh(D) → Kh(D): For every resolution of D, we place a dot on
the identity cobordism on the component that p lies on. Graphically, we represent this
morphism by a dot placed on the diagram at p. Up to sign (and homotopy), this morphism
only depends on the component of the link that p lies on, as can be shown by the explicit
homotopy below.
d=

xp +xq = +

xp +xq = +


/
u
uu
uu
h= uuuu
uu
uu
uu

 zuuu d=
/

xp +xq = +

Next we define the s-invariant. Let FC be a filtered chain complex with an increasing
a
a+1
b
filtration 0 = F C ⊆ F
C ⊆ · · · ⊆ F C = C (this is the opposite convention from [38]
15

since we fixed the dot to be of degree 2 rather than −2). This filtration induces a filtration
i ∗
i
on the homology of C by setting F H (C) = {[v] | v ∈ F C}, whose associated graded
complex is the E∞ -page of the spectral sequence associated to the filtration. Given a non∗
trivial homology class α ∈ H (C), s(α) is the grading of the image of α in the associated
i
graded complex. More explicitly, s(α) = sC (α) = min{i ∈ Z | ∃v ∈ F C : [v] = α}. We
also set s(0) = −∞.
˜ ˜ ˜
˜
˜
Let R be a particular ring with distinguished elements h, t ∈ R. We endow R with an R˜
˜
˜
˜
module structure, where h acts by h and t acts by t and define Kh˜ ˜(D; R) = Kh(D) ⊗R R.
h,t
˜
The grading on Kh(D) induces a filtration on Kh˜ ˜(D; R). We now restrict to the case
h,t
˜
˜
˜
that D represents a knot diagram. In the case R = Q, h = 0, t = 1, the homology of
˜
Kh˜ ˜ (D; R) is isomorphic to two copies of Q, generated by two classes αmin and αmax
h,h
such that s(αmin ) + 1 = s(αmax ) − 1. This is Rasmussen’s s-invariant [38], which we denote
˜
˜
˜
by s(D). Similarly, we can set R = Z2 , h = 1, t = 0. Turner [41] shows that the homology
of Kh(D) is generated over Z2 by two generators αmin and αmax . The discussion in the
last section of [25] applies over any field, thus we can define a “Z2 -Rasmussen invariant” by
s2 (D) := s(αmin ) + 1 = s(αmax ) − 1.

2.4

Khovanov-Rozansky homology

sl(n) homology, defined by Khovanov and Rozansky in [21], is a categorification of the sl(n)
polynomial, a certain specialization of the HOMFLY-PT polynomial that can be obtained
from the fundamental n-dimensional representation of Uq (sl(n)). As noted implicitly by
Gornik [11] and later used by Rasmussen in [37] (see also Krasner [19] and Wu [48]), the
definitions make sense in a more general context: To any polynomial p ∈ Q[x], one can assign
16

a homology theory that conjecturally only depends on the multiplicities of the (complex)
1 n+1 . For
roots of it derivative p (x). sl(n) homology is recovered by setting p(x) = n+1 x
odd n, we establish invariance under positive mutation, that is mutation that respects the
orientations of both 2-tangles involved in it.
Our definitions closely follow [37], but note that we work with Z2 -graded matrix factorizations instead of Z-graded ones in order to get a stronger version of invariance under
Reidemeister moves.
A matrix factorization over a commutative ring R with potential w ∈ R is a free Z2 ∗
2
graded module C equipped with a differential d = (d0 , d1 ) such that d = w ·IC . Following
[37], we use the notation
1
1 d / 0
C o
C .
0
d
Morphisms are simply degree-0 maps between matrix factorizations which commute with
the differential. We say that two morphisms of matrix factorizations φ, ψ : C → C are
homotopic if φ − ψ = d h + hdC for some degree-1 homotopy h : C → C . Thus we
C
can define a category of matrix factorizations over R with potential w and morphisms considered up to homotopy. For a graded ring R, whose grading we will call q-grading, we
also introduce a notion of graded matrix factorizations with homogeneous potential w by
0
1
requiring that both d and d be homogeneous of q-degree 1 deg w. Morphisms between
2
graded matrix factorization are required to have q-degree 0, whereas homotopies must have
q-degree − 1 deg w. The corresponding homotopy category of graded matrix factorizations
2
will be denoted by hmf w (R). For the two different gradings in hmf w (R) we introduce two
types of grading shifts: A shift in the Z2 -grading coming from matrix factorizations will be
denoted by · and a shift in q-grading by {·}. We follow the convention that R{n} has a
17

single generator in q-degree n. Note that if φ : A → B has q-degree d, then the q-degree of
φ : A{kA } → B{kB } is d + kB − kA .

An important class of matrix factorizations is the class of Koszul factorizations, which we
will briefly describe here. For a more detailed treatment, we refer the reader to Section 2.2
of [23] (but note that we switched the order of the arguments of K in order to be consistent
with [22] and [37]). If u, v ∈ R, then K(u; v) is the factorization

R

deg v−deg u
2
o

v /
R
u

We will sometimes write KR (u; v) to clarify which ring we are working over. For u =
T
T
n
(u1 , . . . , un ) , v = (v1 , . . . , vn ) we define K(u, v) = k=1 K(uk ; vk ). This is a matrix
n
factorization with potential k=1 uk vk . We will also use the notation


 u1 v1 




.
.
.
.
K(u, v) =
. 
 .




un vn
If we are not interested in u, we may apply arbitrary row transformations to v: for an
−1 t
invertible matrix X, K(u, v) ∼ K (X ) u, Xv . We describe order-two Koszul matrix
=
factorizations explicitly, thereby fixing a sign convention for the tensor product of matrix
factorizations:
u2

u1 v1
u2 v2

u1
v1 −v2
/
R{k1 + k2 } ⊕ R
= R{k1 } ⊕ R{k2 } o
v2 u1
v1 −u2
18

c

d

c

d

c

d

c

d

a

b

a

b

a

b

a

b

D+

D−

Dr

Ds

Figure 2.3: Positive crossing, negative crossing, oriented smoothing and singular crossing.
The dotted line connecting the two arcs of the oriented smoothing illustrates that we consider
both arcs to be on the same component of the smoothing.
Here k1 =

deg v2 −deg u2
deg v1 −deg u1
deg w
deg w
= deg v1 − 2 and k2 =
= deg v2 − 2 .
2
2

Our definition of Khovanov-Rozansky homology closely follows Rasmussen [37], whose
definitions we amend slightly for technical reasons. We also restrict ourselves to connected
diagrams. To any diagram of a (possibly singular) oriented tangle, which we allow to contain
any of the diagrams depicted in Figure 2.3 as subdiagrams, Rasmussen defines two rings,
which depend only on the underlying 4-valent graph obtained by replacing all of those
diagrams by a vertex. The edge ring R(D) is the polynomial ring over Q generated by
variables xe , where e runs over all edges of the diagram, subject to a relation of the form
xa + xb − xc − xd for each vertex of the underlying 4-valent graph. By setting deg xe = 2
for each edge e of D, R(D) becomes a graded ring. The external ring Rext (D) (called Re
in [37]) is the subring of R(D) generated by the variables associated to the endpoints of
D. Lemma 2.5 in [37] shows that if we associate the variables xi (i ∈ {1, 2, . . . , k}) to the
incoming edges of D and yi to the outgoing edges, then Rext (D) ∼ Q[x1 , . . . , xn , y1 , . . . , yn ]/
=
i yi −

i xi .

Fix a polynomial p ∈ Q[x]. If p is not homogeneous, we will disregard q-gradings below.
To each tangle diagram D, we associate a complex Cp (D) of matrix factorizations over
b
R(D), which we consider to be an object of the category K (hmf w (Rext (D))), where
b
K (C) denotes the homotopy category of bounded complexes over the additive category C
19

and w =

i p(yi ) −

i p(xi ) where xi and yi are associated to the incoming and outgoing

edges as above. Cp (D) is first defined on the diagrams shown in Figure 2.3. In each case
R := R(D) = Rext (D) = Q[xa , xb , xc , xd ]/(xa + xb = xc + xd ). We set

Cp (Dr ) = K(xc − xa ; ∗) 1 = K(∗; xc − xa ){n − 1},
Cp (Ds ) = K(∗; xc xd − xa xb ){−1},
d+
Cp (D+ ) = K(∗; xc xd − xa xb )[−1] − → K(∗; xc − xa ) and
−
d−
Cp (D− ) = K(∗; xc − xa ) − → K(∗; xc xd − xa xb ){−2}[1].
−

Here ∗ is of course determined by the potential in each case; we postpone the definitions of
d+ and d− until we need them in Lemma 3.3.7.
This definition is extended to arbitrary tangle diagrams by the formula

Cp (D) =
i

Cp (Di ) ⊗R(D ) R(D),
i

(2.1)

where Di runs over all crossings of D and the big tensor product is taken over R(D). As
indicated above, we usually view Cp (D) as a matrix factorization over the smaller ring
Rext (D).
Rasmussen shows (Lemma 2.8 in [37])

Proposition 2.4.1. If D is obtained from D1 and D2 by taking their disjoint union and
identifying external edges labeled x1 , . . . , xk in both diagrams, then

R(D) ∼ R(D1 ) ⊗Q[x ,...,x ] R(D2 ) and
=
1
k
20

Cp (D) ∼ Cp (D1 ) ⊗Q[x ,...,x ] Cp (D2 ).
=
1
k
To define reduced Khovanov-Rozansky homology of a link with respect to a marked
component, we pick an edge on the marked component, which we label by x. We view
b
Cp (D) as a complex of matrix factorization over Q[x], i.e. as an object of K (hmf 0 (Q[x])).
◦
Alternatively, we may consider the diagram D obtained from D by cutting it open at the
◦
marked edge. Let x and y be the labels of the incoming and outgoing edge of D , respectively.
◦
Then Cp (D ) is a complex of matrix factorization with potential p(y)−p(x) = 0 over the ring
◦
◦
b
Rext (D ) = Q[x, y]/(y − x) ∼ Q[x] and Cp (D) ∼ Cp (D ) as objects of K (hmf 0 (Q[x])).
=
=
Define the reduced complex

◦
b
Cp (D) := Cp (D )/(x) ∈ K (hmf 0 (Q))

(we use · rather than ¯ in order to avoid confusion with the involution ¯ to be defined later).
·
·
Since we are working over a field and matrix factorizations with potential 0 are simply
Z2 -graded chain complexes, the category hmf 0 (Q) is equivalent to the category of Z2 ⊕ Zb
graded Q-vector spaces by Proposition 2.4.4 below. Hence the category K (hmf 0 (Q) is
b
equivalent to the category K (Z2 ⊕ Z-graded Q-vector spaces), which in turn is equivalent
to the category of Z2 ⊕ Z ⊕ Z-graded Q-vector spaces (bounded with respect to the second
Z summand) by Proposition 2.4.3. Reduced Khovanov-Rozansky homology is the image of
Cp (D){(n − 1)w} under this equivalence of categories, where w is the writhe of D. Since
the theory, is supported in only one of the two Z2 -gradings, we may view it as a bigraded
i,j
vector space. We use the notation Hp (D) for the subspace of this vector space in q-degree
i and homological degree j.

21

Proposition 2.4.2. The definition of reduced sl(n) homology above is equivalent to Khovanov and Rozansky’s original definition in [21].
Proof. Other than working with Z2 -graded matrix factorization rather than Z-graded ones,
our definitions coincide with Rasmussen’s in [37]. Equivalence to the the original formulation
thus follows directly from Proposition 3.12 in [37].
The following two propositions are well-known in the finitely generated case. We verify
that proof carries over to the infinitely generated setting.
k+1
k
k−1
d
k+1 d
k d
− − → . . . be a chain complex over
−−
−
Proposition 2.4.3. Let C = . . . − − → C − → C
−−
Q (with not necessarily finitely generated chain groups). Then C is homotopy equivalent to
a complex with zero differential (its cohomology).
k :=
k
k+1 :=
k
ker(d ) and B
im(d ). Since vector spaces are free as
k
k+1
k
k
k d
−
→ 0 and 0 → B →
modules, the short exact sequences 0 → Z → C − → B
Proof. As usual, let Z

k
k
k
Z → H (C) → 0 split. It is easy to check that with respect to the decomposition C ∼
=
k
k+1
k
k+1 ∼ k
, C decomposes as a direct sum of chain complexes
Z ⊕B
= B ⊕ H (C) ⊕ B
k
k
k id
−
0 → H (C) → 0 and 0 → B −→ B → 0. The Proposition now follows from the fact that
the latter chain complex is homotopy equivalent to the zero complex.
1
1 d / 0
Proposition 2.4.4. Any Z2 -graded chain complex C o
C is homotopy equivalent to
0
d
a Z2 -graded chain complex with zero differential.
Proof. Arguing as in the proof of the previous Proposition, we may decompose C as a direct
0
1 0 / 1
0 id / 0
H (C) , B o
B and B o
B , where the latter
0
id
two complexes are homotopy equivalent to zero complexes.

1
sum of H (C) o

/

0, 0o

/

22

Chapter 3
Link Homology and Mutation
Understanding the behavior of Khovanov homology under Conway mutation has been an
active area of study. Wehrli [44] demonstrated that unlike the Jones polynomial, Khovanov
homology detects mutation of links. Bar-Natan [2] showed that for a pair of mutant knots
(or, more generally, two links that are related by component-preserving mutation) there are
two spectral sequences with identical E2 pages converging to the Khovanov homologies of the
knots. Champanerkar and Kofman [8] relate Khovanov homology to a (mutation-invariant)
matroid obtained from the Tait graph of a knot diagram. The question remains open, but
with coefficients in Z2 it was solved independently by Bloom [6] and Wehrli [45]. In fact,
Bloom proves the more general result that odd Khovanov homology (see Ozsv´th, Rasmussen
a
and Szab´ [36]) is invariant under arbitrary mutation of links. A similar statement cannot
o
hold for the original Khovanov homology, as we know from Wehrli’s example in [44]. Recently, Kronheimer, Mrowka and Ruberman [13] showed that the total rank of instanton
knot homology is invariant under genus-2 mutation, which implies invariance under Conway
mutation.
23

In this chapter, we prove two mutation invariance results about the previously defined
link homology theories. First, we show mutation invariance of the universal sl(2) homology
of knots with coefficients in Z2 , which immediately implies invariance of a variant of Rasmussen’s s-invariant. We then show that reduced sl(n) homology is invariant under positive
mutation if n is odd.
sl(3) homology can be defined over the integers using webs and foams ([26]) and there is
a universal theory similar to universal sl(2) homology ([33]). This theory is invariant under
arbitrary mutation when working with coefficients in Z2 [a, b, c] and a reduced version of the
theory is invariant under positive mutation when working over Z[b]. This can be shown using
similar methods, but we will not discuss the case in this thesis. For n > 3, sl(n) homology
has only been defined over the rational numbers. If a suitable definition over Z is found, we
expect the observed pattern to continue: sl(n) homology should be invariant when working
over Z2 and a reduced version should be invariant under positive mutation for odd n.

3.1

Lemma about invariance of mapping cones

The following lemma is at the heart of the proof. We will use it to show that invariance under
mutation is essentially a property of the category associated to 2-tangles. The functors F
and G are necessary to account for grading shifts; we suggest that the reader think of them
as identity functors and of f as a natural transformation in the center of the category.
Lemma 3.1.1. Let C be an additive category and let F, G and ¯ be additive endofunctors of
·
C, where ¯ : C → C is required to be the identity on objects and an involution on morphisms.
·
Furthermore, let f : F ⇒ G be a natural transformation from F to G and let ∂ : C(A, B) →
C(GA, FB) be an operation defined on the Hom-sets of C with the following properties
24

(1) ∂ is Z-linear, i.e. for φ, ψ ∈ C(A, B), ∂(φ − ψ) = ∂φ − ∂ψ.
¯
¯
(2) For φ ∈ C(A, B), G(φ − φ) = fB ∂φ and F(φ − φ) = ∂φ fA .
(3) Composable morphisms φ ∈ C(A, B) and ψ ∈ C(B, C) satisfy a perturbed Leibniz rule:
¯
∂(ψ φ) = ∂ψ Gφ + F ψ ∂φ.
If C is a chain complex over C with differential d, then f gives rise to a chain morphism
¯
fC : FC → GC. Let C be the chain complex obtained by applying ¯ to the differential of C.
·
Then the mapping cones Cone(fC ) and Cone(fC ) are isomorphic.
¯
Remark 3.1.2. To get a more symmetric statement, we could add the condition ∂(ψ φ) =
¯
∂ψ G φ + Fψ ∂φ to (3). This is unnecessary, however, since this condition follows from the
¯
¯
¯
¯
other conditions by (∂ψ Gφ + F ψ ∂φ) − (∂ψ G φ + Fψ ∂φ) = ∂ψ G(φ − φ) − F(ψ − ψ) ∂φ =
∂ψ f ∂φ − ∂ψ f ∂φ = 0.
Proof. We adopt the following conventions for the mapping cone. For any chain complex
A, let εA : A → A be the identity in even homological heights and the negative of the
identity in odd heights. Note that ε commutes with morphisms of even homological degree
and anti-commutes with morphisms of odd degree. Then the mapping cone Cone(fC ) is
Fd
. Since Fd here has degree 1, it
given by FC[−1] ⊕ GC with differential
fC ◦ εFC Gd
is easy to check that this defines a differential.
We claim that the horizontal arrows in Figure 3.1 give rise to an isomorphism between Cone(fC ) and Cone(fC ).
¯

I ∂d ε
I

is invertible with inverse

it remains to check that it defines a chain morphism, i.e. that
I ∂d ε
I

Fd
f ε Gd

¯
Fd
¯
f ε Gd

I −∂d ε
I
I ∂d ε
I

, so
=

. Since Fd : FC[−1] → FC[−1], Gd : GC[−1] → GC[−1] and
25

FC[−1] ⊕ GC

I ∂d ◦ εGC
I

O

/

FC[−1] ⊕ GC
O

¯
Fd
¯
f C ◦ εF C G d
¯
¯

Fd
fC ◦ εFC Gd
FC[−1] ⊕ GC

/

I ∂d ◦ εGC
I

FC[−1] ⊕ GC

Figure 3.1: Proof of Lemma 3.1.1
fC : FC[−1] → GC each have homological degree −1 as they are part of the differential on
Cone fC , they anti-commute with ε. Similarly, ∂d : GC → FC[−1] has homological degree
0 as it is part of the chain morphism, so it commutes with ε.
¯
¯
¯
The claim now follows from F d = Fd − ∂d f = Fd + ∂d εf ε, from F d ∂d ε = F d ∂d ε −
2
¯
¯
¯
∂(d ) ε = F d ∂d ε − ∂d Gd ε − F d ∂d ε = −∂d Gd ε = ∂d ε Gd and from f ε∂d ε + G d =
¯
f ∂d + G d = Gd.

3.2

Mutation invariance of sl(2) homology

In this section, we show that universal sl(2) homology over the ring Z2 [h, t] is invariant
under mutation of knots. This generalizes an earlier result of Wehrli [45], who considered (in
our notation) the case h = 0. It also generalizes a special case of Bloom’s proof of mutation
invariance of odd Khovanov homology [6]. An advantage of working in this generality is that
we immediately obtain that the Z2 -Rasmussen invariant is preserved by mutation. We do
not know whether this invariant is identical to the invariant defined by Rasmussen in [38],
but we expect that it shares all its properties, in particular that it gives a lower bound on
the slice genus.
26

Since we work in characteristic 2 throughout the section, let us define R2 := R ⊗ Z2 ∼
=
b
Z2 [h, t] and Kh2 (D) := Kh(D)⊗Z2 , which we may view as an object of K (mKob(E)⊗Z2 )
b
or K (mKob (E) ⊗ Z2 ). We will prove that if D and D are two knot diagrams related by
mutation, then Kh2 (D) ∼ Kh2 (D ).
=
Let Kob(4) = Kob(E), Kob (4) = Kob (E), mKob(4) = mKob(E) and mKob (4) =
mKob (E) where E is the set of four points on the boundary of the unit disc in north-east,
north-west, south-west and south-east position. We will describe the category Kob (4)
explicitly.

Proposition 3.2.1. The Z[h, t]-gmod-category Kob (4) has objects

and

and morphism

spaces

Kob (4)( , ) = Z[h, t]
Kob (4)( , ) = Z[h, t]

deg
:

= deg
→

and

,

Kob (4)( , ) = Z[h, t]

= deg

,

Kob (4)( , ) = Z[h, t]

Here deg

, , ,

, , ,

= 0, deg

= 1 and deg
:

→

= deg
= deg

= deg

= deg

= 2, deg

= deg

= 4,

= 3. The identity morphisms of Kob (4) are

. In addition, the following relations hold:

◦

(3.1)

◦

=h +t

(3.2)

◦
27

=h +t

=h +t

(3.3)

◦

=h +t

(3.4)

◦

=

◦

=

(3.5)

◦

=

◦

=

(3.6)

◦

=

◦

=

◦

=

◦

=

(3.7)

◦

=

◦

=

◦

=

◦

=

(3.8)

◦

=

+

−h

(3.9)

◦

=

+

−h

(3.10)
(3.11)

Proof. Let α be a morphism in Kob (4). We can represent α as a linear combination of
cobordisms without closed components. By the neck-cutting relation, we can further reduce
α to a linear combination of dotted cobordisms of genus 0. If the source and the target of
α are identical, we can additionally represent it as a linear combination of dotted sheets.
Finally, if there is a component with more than one dot on it, we can use the two-dot relation
to reduce the number of dots. Thus α can be represented as a linear combination of the
cobordisms listed above.
Relations (3.1) through (3.4) now are consequences of the two-dot relation, (3.5) through
(3.8) follow from isotopy and (3.9) and (3.10) come from the neck-cutting relation. One
readily checks that these relations determine composition completely. It can be seen from
closing up the tangle in the two possible ways that there are no non-trivial relations between
the generators of the category.

Corollary 3.2.2. Kob(4) is isomorphic to the free Z[h]-gmod-category generated by
28

,

,

,

,

and

modulo the ideal generated by relations (3.1)−(3.2), (3.3)−(3.4), (3.9),

(3.10) and the left hand sides of equations (3.5) through (3.8).

Proof. It is clear that the relations in Proposition 3.2.1 imply those in the corollary, we must
show that the converse is true. Define t :
t:

→

by t =

◦

−h

=

◦

→

by t =

◦

−h

=

◦

−h

and

− h . In order to prove the corollary, we need to

show that t is in the center of the category. We will show that t commutes with

below;

commutativity with the other generators follows similarly.

t◦

=

◦

◦

−h ◦

=

◦

◦

−h ◦

=

◦t

Given a category described by generators and relations as above, we would like to define
an operation ∂ by specifying its value on generators and then extending via the perturbed
Leibniz rule from Lemma 3.1.1(3). We have to make sure this yields a well-defined operation.

Lemma 3.2.3. Let A be an R-gmod-category given by generators and relations, which we
˜
˜
write as A = A/I, where A is the free R-gmod-category corresponding to the generators and
I is the homogeneous ideal generated by the relations. For any generator φ from A to B,
¯
˜
˜
which we view as an element of A(A, B), fix a value φ ∈ A(A, B) of the same degree and a
˜
˜
value ∂φ ∈ A(A, B) of degree deg φ − 2. ¯ extends uniquely to a R-gmod-functor ¯ : A → A,
·
· ˜
which is the identity on objects. We require that ¯ be an involution.
·
¯
(1) There is a unique R-linear extension of ∂ (of degree −2) satisfying ∂(ψ φ) = ∂ψ φ + ψ ∂φ
(compare Lemma 3.1.1(3)).
29

(2) If for all relations r, both r and ∂r lie in I, then for any morphism ψ ∈ I, ∂ψ ∈ I.
¯
Hence ∂ descends to a well-defined operation on A.
¯
(3) For all objects A of A, choose an element fA of A(A, A) of degree 2 such that φ − φ =
fB ∂φ = ∂φ fA holds for every generator φ : A → B. Let C = Mat(A0 ) and define
F : C → C to be the shift functor {2} and G : C → C the identity functor. Then f gives
rise to a natural transformation f : F ⇒ G and the category C together with ¯ and ∂
·
satisfies all hypotheses of Lemma 3.1.1.

Proof. (1) We first note that if 1A is the identity morphism on an object A (i.e. the generator
˜
of the free graded R-module A(A, A) corresponding to the empty path), then ∂1A = ∂(1A ◦
¯
1A ) = ∂1A ◦ 1A + 1A ◦ ∂1A = 2∂1A (note that ¯A = 1A since ¯ is a functor), hence
1
·
˜
∂1A = 0. We can use R-bilinearity and the formula given for ∂(ψ φ) to define ∂ on A. To
show well-definedness, one readily checks that the formula for ∂(ψ φ) trivially holds if ψ or
φ are identity morphisms and that for the composition of three morphisms φψρ,

∂((ρψ)φ) = ∂(ρ(ψφ)) = ∂ρ ψ φ + ρ ∂ψ φ + ρ ψ ∂φ.
¯
¯¯

(2) Every element in I can be written as the sum of elements of the form φψρ, where
ψ is a relation. It follows from the computation in (1) that ∂(φψρ) ∈ I, hence I is closed
under ∂.
(3) We constructed φ to satisfy conditions (1) and (3) of Lemma 3.1.1. Condition (2)
holds for generators by the assumption and can be extended to arbitrary morphisms of
¯
¯
¯
C: If f ∂φ = G(φ − φ) and f ∂ψ = G(ψ − ψ), then f ∂(ψ φ) = f ∂ψ Gφ + f F ψ ∂φ =
¯
¯
¯
¯
¯¯
f ∂ψ Gφ + G ψ f ∂φ = G(ψ − ψ) Gφ + G ψ G(φ − φ) = G(ψ φ − ψ φ). In the same way, one can
30

¯
show that ∂φ f = F(φ − φ).

We now define a differential on Kob(4) ⊗ Z2 of degree −2 by setting ∂
= ∂

∂

=

, ∂

=

=

,

= 0. We do not specify ¯ yet, but we require that
·

= 0 and ∂

{ , } = { , }, { , } = { , },
f

=∂

=

and

=

. We set f

=

+

and

+ .

We are now ready to verify the hypotheses of Lemma 3.2.3. For every relation r mentioned
in Corollary 3.2.2, we must show that r and ∂r are zero when viewed as morphisms in
¯
Kob(4) ⊗ Z2 . It is immediately clear that this is the case for r. The relations (3.1)−(3.2),
¯
(3.5) and (3.9) can be written as r1 =
◦

r3 =

+

◦α+β◦

+

◦

+

◦

+h

+

◦

+

◦

,

+ h , respectively; (3.7) is equivalent to the set of relations rα,β =

where α ∈ { , } and β ∈ { , }.

One readily computes that in Kob(4) ⊗ Z2 , ∂r1 =
+

+ h , r2 =

+

= 0, ∂r3 = 0 + 2 + 0 = 0 and ∂rα,β =

+
+

+

+

+ 2h = 0, ∂r2 =

= 0.

¯
We will also verify that for all generators φ ∈ Kob(4) (A, B), φ − φ = fB ∂φ = ∂φ fA .
This is clear for φ =
,

or

and φ =

¯
since in these cases, φ = φ and ∂φ = 0. If φ is one of

,

¯
, then ∂φ = 1 and our definition of f implies that φ − φ = f .

Mutation of the inner tangle is given by one of the involutions Rx , Ry and Rz , where
Rx and Ry are reflections along the x-axis and y-axis, respectively and Rz = Ry ◦ Rx is
◦
rotation by 180 around the origin. Rx , Ry and Rz induce functors on Kob(4), Kob (4),
mKob(4) and mKob (4), which we all denote by Rx , Ry and Rz . It is easy to see that
these functors commute with the delooping functor mKob(4) → mKob (4) (compare [45]).
On Kob (4), the functors Rx , Ry and Rz are identity functors on objects and are given
by Rx

= Ry

= Rz

=

, Rx

= Ry
31

= Rz

=

, Rx

= Ry

= Rz

=

,

Rx

= Ry

= Rz

=

, Rx

= Ry

= Rz

=

and Rx

= Ry

= Rz

=

on

generating morphisms. Thus they satisfy all the conditions we imposed on ¯ and ∂ above, so
·
Lemma 3.1.1 applies and

¯
¯
Cone f : Kh2 (D){2} → Kh2 (D) ∼ Cone f : Kh2 (D){2} → Kh2 (D) .
=

We have only shown this isomorphism over Z2 [h] so far. To show that it holds over R2 we
must verify that the isomorphism commutes with t. But this follows immediately from the
fact that t is in the center of the category as shown in the proof of Corollary 3.2.2.
b
Before we can prove the theorem we need to ensure that the category K (R2 -fgmod) is
well-behaved.

Definition 3.2.4. An additive category C is called Krull-Schmidt if every object of C is
the direct sum of finitely many indecomposable objects and if

∼
i∈I Xi =

j∈J Yj for

indecomposable objects Xi and Yj implies that there is a bijection φ : I → J such that
Xi ∼ Yφ(i) .
=

b
Proposition 3.2.5. The category K (R2 -fgmod) is Krull-Schmidt.

b
Proof. We first show that idempotents in K (R2 -fgmod) split. Let e : C → C be such an
idempotent. Proposition 3.2 in [7] shows that idempotents split in any triangulated category
that has arbitrary direct sums (in particular, infinite direct sums). We cannot use this
statement directly, since we need to stay in a category of complexes of finitely generated
modules. However, we may use their construction to define a complex Ce as a totalization
32

of the complex of complexes

1+e
e
1+e
···C − − C −− C − − C
−→
−→
−→

by successively taking appropriate mapping cones. It is easy to see that Ce is a complex of
finitely generated free graded R2 -modules that is bounded above but not bounded below.
Ce gives rise to a splitting of e, i.e. there are maps r : C → Ce and i : Ce → C such that
r◦i

id and i ◦ r

e. The homology of Ce is supported in finitely many degrees, say

k
H (Ce ) ∼ 0 for k ≤ N . Write Ce as
=
n−1
n−2
n
n−1 d
n−2 d
− − → Ce
−−
− − → Ce
−−
· · · Ce

and truncate it to get a complex Ce given by
n−1
N
n
n−1 d
N
N d
− − → Ce ,
−−
−→
ker d −→ Ce − − · · · Ce
−

which is quasi-isomorphic to Ce via the evident quasi-isomorphism Ce → Ce . Since R2 -gmod
−
−
has enough projectives, the functor K (R2 -fgmod) → D (R2 -gmod) from the bounded
above homotopy category of projectives to the bounded above derived category is an equiva−
lence of categories. Hence Ce and Ce are isomorphic in K (R2 -fgmod) and maps r : C →
Ce and i : Ce → C with r ◦ i

id and i ◦ r

e give rise to a splitting of e.

b
Next we show that the endomorphism rings of indecomposable objects of K (R2 -fgmod)
are local. As a Z2 -vector space, End(C) is finite-dimensional since for all k and l, the
space R2 -fgmod(R2 {k}, R2 {l})) is finite-dimensional. If x ∈ End(C), then there are natural
33

numbers m < n such that x
x

m(n−m−1)+m

n

implies that x

m
2m(n−m)
m(n−m−1)+m+m(n−m)
= x . Thus x
= x
=
m(n−m)

is an idempotent, hence it is either an isomorphism
k
k
2
2
or 0. In the first case, x is a unit in End(C); in the second case, (1 + x) = 1 + x = 1
k
for 2 ≥ m(n − m), so 1 + x is a unit.
Finally, we note that End(C) is finite-dimensional for any object C, hence any object
admits a decomposition into finitely many indecomposables. The standard proof of the
Krull-Schmidt theorem now applies (see for example Theorem 7.5 in [27]).

Lemma 3.2.6. Let C be a Krull-Schmidt category and F : C → C be an autofunctor of C
2
such that F X

X for all non-zero objects X ∈ C. Then X ⊕ FX ∼ Y ⊕ FY implies that
=

X ∼Y.
=

Proof. X and Y split into the same number n of indecomposable objects; we prove the
lemma by induction on n. If X is the zero object, then so is Y and hence X ∼ Y . Now
=
suppose the lemma holds for all n < n. Write X =
assumption

i Xi ⊕

∼
i FXi =

i Yi ⊕

n
i=1 Xi and Y =

n
i=1 Yi , so by

∼
i FYi . We claim that Xj = Yk for some j and k.

Otherwise, each Xi would have to be isomorphic to one FYi and each Yi isomorphic to one
2
FXi , which would imply X ∼ FY and Y ∼ FX and therefore X ∼ F X, a contradiction.
=
=
=
Write X ∼ Xj ⊕ X and Y ∼ Yk ⊕ Y in the obvious way and notice that by induction
=
=
hypothesis, X ∼ Y , hence X ∼ Y .
=
=

Theorem 3.2.7. Let K and K be two knots related by mutation. Specifically, let D and
¯
D be 2-tangles such that D ∪ D represents K and D ∪ D represents K . Then Kh2 (K) ∼
=
Kh2 (K ).
34

Proof. By the discussion above,

Cone f : Kh2 (D){2} → Kh2 (D)
∼ Cone f : Kh (D){2} → Kh (D)
¯
¯
=
2
2

Tensoring both sides with Kh2 (D ) we obtain an isomorphism

Cone f ⊗ 1 : Kh2 (K){2} → Kh2 (K)
∼ Cone f ⊗ 1 : Kh (K ){2} → Kh (K )
=
2
2

Since f ⊗ 1 is the difference xp − xq , where p and q are opposite endpoints of the tangle
D. After closing up p and q lie on the same component, hence f ⊗ 1 is null-homotopic. It
follows that

Kh2 (K)[−1]{2} ⊕ Kh2 (K) ∼ Kh2 (K )[−1]{2} ⊕ Kh2 (K )
=

Applying Lemma 3.2.6 with FA = A[−1]{2}, we see that Kh2 (K) ∼ Kh2 (K ).
=
Corollary 3.2.8. The Z2 -Rasmussen invariant s2 is invariant under mutation of knots.
Proof. It is clear that the definition of s2 (K) only depends on the homotopy type of the
complex Kh2 (K).

3.3

Mutation invariance of sl(n) homology

This section is devoted to proving the following result.
35

Theorem 3.3.1. If L and L are two links related by component-preserving positive mutation
and n is odd, then their reduced sl(n) homologies are isomorphic (reduced with respect to
the component of the mutation, defined after Definition 2.1.2). More generally, let p(x) =
k a2k x

2k

be a polynomial with only even powers of x, then the reduced Khovanov-Rozansky

homologies of L and L associated to this polynomial are isomorphic.
Using Rasmussen’s spectral sequence from HOMFLY-PT homology to sl(n) homology,
and the fact that HOMFLY-PT homology of knots is finite-dimensional, we get the following
corollary.
Corollary 3.3.2. If K and K are two knots related by positive mutation, their HOMFLYPT homologies are isomorphic.
We prove the theorem by first showing that the Khovanov-Rozansky complex of the inner
2-tangle can be built out of the complexes assigned to two basic diagrams: a pair of arcs and
a singular crossing. For the Khovanov-Rozansky complex, we follow Rasmussen’s definitions
from [37], since Khovanov and Rozansky’s original definitions are not general enough to
serve our purpose. We then derive a criterion for a certain mapping cone of this complex
to be invariant under reflection, which turns out to be the case for odd n in the case of
positive mutation. Closing up the tangle, we see that the mapping cone computes reduced
Khovanov-Rozansky homology.
Finally, we note that our result is consistent with calculations for the Kinoshita-Terasaka
knot and the Conway knot carried out by Mackaay and Vaz [34].

We quickly summarize the organization of the section. In subsection 3.3.1, we reduce the
problem to the case of mutation of a 2-tangle in what we call braid form. In subsection 3.3.2,
36

we investigate how the Khovanov-Rozansky complex behaves under positive mutation. In
subsection 3.3.3, we show how to represent the Khovanov-Rozansky complex of a 2-tangle in
braid-form as a complex over a particularly simple category. In subsection 3.3.4, we derive
a general criterion for when a chain complex over an additive category is isomorphic to its
image under a certain involution functor and show how this criterion applies to the problem
at hand. In subsection 3.3.5, we combine the results from the previous subsections to prove
Theorem 3.3.1.

3.3.1

Topological considerations

In this subsection, we show that we may assume that the inner tangle is presented in a
specific form.
Definition 3.3.3. We say that a 2-tangle is in braid form if it is represented in the following
way, where the rectangle represents an open braid.

···

Theorem 3.3.4. Let L be an oriented link and L be a mutant of L obtained by positive
mutation. Then the mutation can be represented on a diagram whose inner tangle is given
in braid form by a transformation of type Ry .
The following two lemmas immediately imply the Theorem.
Lemma 3.3.5. We may assume that the endpoints of the inner tangle are oriented as in
Figure 3.2(a) and that the transformation of the inner tangle is of type Ry .
37

c

d

c

d

a

b

a

b

(a)

(b)

Figure 3.2: The two possible orientations of the endpoints of inner tangle

R

≈

R

≈

R

R

Figure 3.3: Rz mutation on a tangle of type (b) is equivalent to Ry mutation on a tangle of
type (a)
Lemma 3.3.6. Any 2-tangle with endpoints oriented as in Figure 3.2(a) can be represented
by a diagram in braid form.

Proof (of Lemma 3.3.5). If the tangle has two adjacent endpoints with the same orientation,
it is isotopic to a tangle with endpoints as depicted in Figure 3.2(a) and the only positive
mutation is of type Ry . Otherwise we are in case (b) of Figure 3.2, where the only positive
mutation is of type Rz . But we can realize this type of mutation by Ry -mutation on a tangle
of type (a), as illustrated in Figure 3.3.

Proof (of Lemma 3.3.6). The proof uses a slight variation of the Yamada-Vogel [43, 49] algorithm to prove an analog of Alexander’s Theorem for 2-tangles. We follow Birman and
Brendle’s exposition of the proof [5].
38

c

d

α

c
β

a

d

α

b

β
a

(a)

b

(b)

Figure 3.4: A closure of the tangle and its Seifert picture

···

···

···

···

Figure 3.5: Transforming the diagram into braid form

Close the tangle by two arcs α from c to a and β from d to b as in Figure 3.4(a). The
algorithm works by repeatedly performing a Reidemeister II move in a small neighborhood
of a so-called reducing arc. The algorithm is performed on the Seifert picture of the link
diagram, which is depicted in Figure 3.4(b). A reducing arc is an arc connecting an incoherently oriented pair of Seifert circles that intersects the Seifert picture only at its endpoints.
Since the Seifert circles that α and β belong to are coherently oriented, the the unbounded
region of the Seifert picture in Figure 3.4(b) cannot contain a reducing arc. Hence we may
push the reducing arc into the circle. The algorithm now gives us a tangle diagram whose
Seifert circles and Seifert arcs (from a to c and from b to d) are coherently oriented. This
implies that all Seifert circles lie nested inside each other to the left of the left arc and to
the right of the right arc, in other words it can be represented by a diagram in the form
39

illustrated on the left of Figure 3.5. But this can be easily transformed into braid form, as
seen on the right of Figure 3.5.

3.3.2

Behavior of the Khovanov-Rozansky chain complex under
reflection

¯
Lemma 3.3.7. Let D be an oriented (possibly singular) tangle diagram and D be the reflection of D. Label the endpoints of D by e0 , e1 , . . . , e2k−1 , and the corresponding endpoints
¯
¯
¯
of D by e0 , e1 , . . . , e2k−1 . Then Cp (D) = φ(Cp (D)), where φ : R(D) → R(D) is the ring
˜ ˜
˜
homomorphism given by φ(xe ) = −xe .
˜i
i
Proof. If D is one of the diagrams shown in Figure 2.3, Cp (D) is one of the following
complexes of matrix factorizations.
Cp (D+ ):
R{1 − n} o
O

xc −xb
R{3 − n} o

Cp (D− ):

xc −xa
w
xc −xa

/

(xc −xa )(xc −xb )
w
(xc −xa )(xc −xb )

R
O

R{1 − n} o
O

/

1
R

1
R{1 − n} o

Cp (Dr ):
Ro

xc −xa
w
xc −xa

(xc −xa )(xc −xb )
w
(xc −xa )(xc −xb )

/

R{−2}
O

xc −xa

xc −xb
/

w
xc −xa

R

/

Cp (Ds ):
R{n − 1}

R{2 − n} o

(xc −xa )(xc −xb )
w
(xc −xa )(xc −xb )

/

R{−1}

Note that φ(xc − xa ) = −x ˜ + x˜ = xc − xa , φ(xc − xb ) = −x ˜ + xa = xc − x˜ and
˜
˜
˜
˜
d
b
d
b
2k
2k
2k
2k
2k
2k
2k
2k
φ(w) = φ
˜
˜
k a2k (xc + xd − xa − xb ) =
k a2k (xc + xd − xa − x˜ ) = w.
˜
˜
b
40

Hence all maps in the above diagrams are mapped by φ to the same maps with xa , x˜, xc
˜ b ˜
¯
and x ˜ in place of xa , xb , xc and xd , respectively, that is φ maps Cp (D) to Cp (D).
d
The general case follows from (2.1): It is clear that by taking the internal edges of D
¯
into consideration, we can extend φ to an isomorphism between R(D) and R(D). Hence we
¯
¯
get isomorphisms Cp (Di ) ⊗R(D ) R(D) ∼ Cp (Di ) ⊗R(D ) R(D), which in turn induce an
=
¯
i
i
¯
isomorphism Cp (D) ∼ Cp (D).
=
In light of the Lemma, we will simply denote the homomorphism φ by ¯
·.

3.3.3

Khovanov-Rozansky Homology of 2-tangles

In this subsection, we investigate the Khovanov-Rozansky homology of 2-tangles in braid
form. Denote the variables corresponding to the endpoints a, b, c and d of the tangle by xa ,
xb , xc and xd , respectively. The complex associated to such a tangle is a complex of graded
matrix factorizations over the ring R = Q[xa , xb , xc , xd ]/(xa + xb = xc + xd ) with potential
w = p(xc ) + p(xd ) − p(xa ) − p(xb ). Let hmf 2 denote the full subcategory of hmf w (R)
whose objects are direct sums of shifts of Cp (Dr ) and Cp (Ds ).
Theorem 3.3.8. Let D a connected diagram of a 2-tangle in braid-form. Then Cp (D) is
b
b
isomorphic in K (hmf w (R)) to an object of K (hmf 2 ).
Before proving the theorem, we need to recall an important tool for dealing with matrix
factorizations: ‘excluding a variable’. We quote Theorem 2.2 from [23].
Theorem 3.3.9. Let R be a graded polynomial ring over Q and and u, v ∈ R[y] two polynomials, with v being monic in y. Furthermore, let w ∈ R and M be a graded matrix
¯
factorization over R[y] with potential w = w − uv. Then M/(v) and K(u; v) ⊗ M are iso¯
41

morphic as objects of hmf w (R). We say that we exclude the variable y to obtain M/(v)
from K(u; v) ⊗ M .

The Theorem is only stated for ungraded matrix factorization in [23], but it is trivial to
check that the quotient map K(u; v) ⊗ M → M/(v) constructed in the proof is of degree
0.
We will also need another well-known result about Koszul matrix factorizations; this is,
for example, the n = 2 special case of Theorem 2.1 in [23].

Theorem 3.3.10. Let R be a graded polynomial ring over Q and v1 , v2 ∈ R be relatively
∗ v1
prime. Then any two Koszul matrix factorizations of the form
with the same
∗ v2
potential are isomorphic.

In the same spirit, we show that a matrix factorization that is almost the direct sum of
two order-two Koszul matrix factorizations can be transformed into an honest direct sum.
˙
¨
˙
¨
Theorem 3.3.11. Let R be a graded polynomial ring and R = R{k} and R = R{k} be free
R-modules of rank 1, then any graded matrix factorization of the form

V / ˙
˙
¨
¨
˙
˙
¨
¨
R{ka + kb } ⊕ R ⊕ R{kc + kd } ⊕ R
R{ka } ⊕ R{kb } ⊕ R{kc } ⊕ R{kd } o
U
with



b ∗ 0

a ∗ 0

U =
0 ∗ d

0 ∗ c


∗

∗


∗

∗



and

∗ ∗ ∗

a −b 0

V =
∗ ∗ ∗

0

where gcd(a, b) = gcd(c, d) = 1 and kx = deg x −
42

0

∗




0

,
∗

c −d

deg w
2 for x ∈ {a, b, c, d}, is isomorphic to

a matrix factorization of the form

∗ a
∗ b

∗ c

˙
{k} ⊕

∗ d

¨
{k}

Proof. Let



0 b2


a ∗ 0 a 

2
U =

0 d d ∗ 
1


0 c1 c ∗
b




c2 −d2


 a −b 0
0 


V =
.
a −b
∗
∗ 
1
 1

0
0
c −d

∗

and

∗

∗

Computing the lower left and the upper right quadrant of U V = wI, we see that

d1 d
c1 c

a

−b

a1 −b1

b b2
a a2

= 0 and

c2 −d2
c −d

= 0.

Since gcd(a, b) = gcd(c, d) = 1, the rank of each of these matrices is at least 1, so none of
a

−b

= −b1 a + a1 b and there exists an α ∈ R
a1 −b1
such that a1 = α1 a and b1 = α1 b. Similarly, we can find an α2 ∈ R, such that a2 = α2 a
them can have rank 2. Hence 0 = det

and b2 = α2 b, as well as βi ∈ R (i ∈ {1, 2}) such that ci = βi c and di = βi d. The fact that
the two matrix products above are 0 implies that βi = −αi . We now perform a change of
basis,

˙
˙
¨
¨
R{ka } ⊕ R{kb } ⊕ R{kc } ⊕ R{kd } o

V
/

˙
˙
¨
¨
R{ka + kb } ⊕ R ⊕ R{kc + kd } ⊕ R
O

U

X
˙
˙
¨
¨
R{ka } ⊕ R{kb } ⊕ R{kc } ⊕ R{kd } o

V
U
43

/

−1 X


˙
˙
¨
¨
R{ka + kb } ⊕ R ⊕ R{kc + kd } ⊕ R

where




X=



1

α2
1
−α1 1





b



a


, U = 

0


1
0

∗ 0 0







∗ ∗ 0 0



a −b 0 0 
∗ 0 0



 and V = 
,
0 0 ∗ ∗ 
0 d ∗



0 c ∗
0 0 c −d

the lower row being exactly the desired direct sum of Koszul matrix factorizations.
We still need to verify that C is of degree 0: We have deg α1 = deg a1 − deg a =
deg w
deg w
¨
˙
¨ ˙
( 2 +ka + k −kc −kd − k)−(ka + 2 ) = k − k −kc −kd and deg α2 = deg a2 −deg a =
deg w
deg w
˙
¨
˙ ¨
˙
( 2 + k − kb − k) − (ka + 2 ) = k − k − ka − kb , which implies deg(−α1 : R →
¨
¨
˙
R{kc + kd }) = deg(α2 : R → R{ka + kb }) = 0.
The following proposition is an analog of Lemma 4.10 and Propositions 4.3–4.6 in [37]
and Lemma 3 and Propositions 4–7 in [22]. Unfortunately, we cannot deduce it from any of
the previous results: The theory introduced in [21] is weaker than what we consider here (in
the sl(2) case, this is the difference between Khovanov Homology and Bar-Natan’s universal
variant [1]). We also cannot use the results in [37], which are only shown to hold up to a
notion of quasi-isomorphism. However, the proofs in Rasmussen’ paper can be modified to
apply to our situation.
Proposition 3.3.12. The following isomorphisms hold in the homotopy category of matrix
factorizations over the external ring corresponding to the diagrams.
(a) Let D be a diagram of a fully resolved tangle, and D be a diagram obtained from D by
replacing a smoothing of type Dr (See Figure 2.3) by a pair of arcs without increasing
the number of components. Then Cp (D) ∼ Cp (D ).
=
(b) Up to grading shifts, Cp (DO ) is isomorphic to a direct sum of n copies of Cp (DA ).
44

c d
DA

c d

Dr

a

Ds
a b

c

a b
c d

c

DO

b

DI

b

a

DIIIa

a

d e f

DII

d e f

x
z

y

DIIIa

a b c
d e f
DIIIb

DIV

DIIIb
a b c

Figure 3.6: Singular braid diagrams

45

y

a b
d e f

a b c
d e f

a b c

x

a b c

(c) Up to grading shifts, Cp (DI ) is isomorphic to a direct sum of n − 1 copies of Cp (DA ).
(d) Up to grading shifts, Cp (DII ) ∼ Cp (Ds ) ⊕ Cp (Ds ).
=
(e) Up to grading shifts, Cp (DIIIa ) ⊕ Cp (DIIIb ) ∼ Cp (DIIIa ) ⊕ Cp (DIIIb ).
=
(f ) Up to grading shifts, Cp (DIV ) ∼ Cp (DIIIb ) ⊕ Cp (DIIIb ).
=
Proof. (a) Since D and D are connected, their external rings Rext (D) and Rext (D ) are
identical. Since R(D ) = R(D)/(xa = xc ) and R(D) ∼ R(D )[x] by Lemma 2.4 in [37], xa
=
and xc are different elements of R(D). If xa and xc were both linear combinations of external
edges, then their difference xc − xa would be a linear combination of external edges as well.
But xc − xa = 0 ∈ R(D) and xc − xa = 0 ∈ R(D ), which contradicts Rext (D) = Rext (D ).
Assume w.l.o.g. that xc is not a linear combination of external edges. Since K(∗; xc − xa )
appears as a factor of Cp (D), we may exclude the variable xc to obtain Cp (D ).
(b) We have R(DO ) = Q[xa , xb , xc ]/(xc − xa ), Rext (DO ) = Q[xa , xc ]/(xc − xa ) and
R(DA ) = Q[xa ], hence

Cp (DO ) = K xc − xa ;

p(xc ) + p(xb ) − p(xa ) − p(xb )
xc − xa

1 = K(0; p (xa ) − p (xb )) 1 .

Excluding the variable xb , we obtain
n−1
Cp (DO ) ∼
=

Cp (DA ) 1 {2i}.
i=0

(c) As in part (b), we have R(Ds ) = Q[xa , xb , xc ]/(xc − xa ), Rext (Ds ) = Q[xa , xc ]/
46

(xc − xa ) and R(DA ) = Q[xa ], so

Cp (DI ) = K

p(xc )+p(xb )−p(xa )−p(xb )
; (xc − xa )(xc − xb ) {−1}
(xc −xa )(xc −xb )

=K

p (xa )−p (xb )
; 0 {−1}
xa −xb

=K

0;

p (xa )−p (xb )
xa −xb

1 {2 − n},

so once again we may exclude xb to get
n−2
Cp (DI ) ∼
=

Cp (DA ) 1 {2 − n + 2i}.
i=0

(d) Choose labels as in Figure 3.6 and set x := xx and y := xy . As matrix factorizations
over R(DII ),

Cp (DII ){2} =
=
∼
=

=

∗ (x − xa )(x − xb )
∗ (xc − x)(xc − y)
∗ (x − xa )(x − xb )
∗ (xc − x)(x − xd )
∗

(x − xa )(x − xb )

∗ (xc − x)(x − xd ) + (x − xa )(x − xb )
∗

(x − xa )(x − xb )

∗ (xc − xa )(xc − xb )

Let R = Rext (DII ) = Rext (Ds ). Excluding the variable x, we get a matrix factorization
2
K(α + βx; (xc − xa )(xc − xb )) over the ring R = R[x]/(x = (xa + xb )x − xa xb ) whose
potential (α + βx)(xc − xa )(xc − xb ) has to lie in R, hence β = 0. As a graded R-module,
47

R ∼ R⊕R{2}, so Cp (DII ) ∼ KR (α; (xc −xa )(xc −xb )){−2}⊕KR (α; (xc −xa )(xc −xb )) ∼
=
=
=
Cp (Ds ){−1} ⊕ Cp (Ds ){1}.

(e) Choose labels as in Figure 3.6, and set x := xx , y := xy and z := xz . Let R =
Rext (DIIIa ), and note that R(DIIIa ) ∼ R[x]. As matrix factorizations over R(DIIIa ),
=


∗ (x − x )(x − z)


d
a d


Cp (DIIIa ){3} = ∗ (xe − x)(xe − y)



∗ (z − x )(z − x ) 

b
c


∗

(xd − xa )(xa − x)




= ∗
(xe − x)(x − xf )



∗ (x + x − x − x )(x + x − x − x )

d
a
b
d
a
c


∗

(xd − xa )(xa − x)




∼ ∗
(xe − x)(x − xf )
,
=




∗ x x + x x + x x − x x − x x − x x 
a b
b c
c a
c d
d e
e f
where the last line is obtained from the previous one by adding the top right and the center
right entry to the bottom right entry. Let p = xe + xf , q = xe xf , α = xd − xa and
β = xa xb + xb xc + xc xa − xc xd − xd xe − xe xf , so that the last line reads


∗
axa − ax 




∼ ∗ −x2 + px − q
Cp (DIIIa ){3} =




∗

b
Using the second row to exclude the variable x, we obtain an order-two Koszul matrix
2
factorization over the ring R = R(DIIIa )/(x = px − q), which is given explicitly (with
respect to the standard decomposition of R as a free R-module of rank two) as Cp (DIIIa ) ∼
=
48

V /
R1 o
R0 , where
U

R1 = R{3 − n} ⊕ R{5 − n} ⊕ R{3 − n} ⊕ R{5 − n},
R0 = R{6 − 2n} ⊕ R{8 − 2n} ⊕ R ⊕ R{2},


β

∗ ∗

0





 0
β
∗ ∗


A=

αx
αq
∗ ∗
 a

−α α(xa − p) ∗ ∗


and

∗

∗

∗

∗



 ∗
∗
∗
∗ 


B=
.
αx
αq
−β 0 
 a

−α α(xa − p) 0 −β

We apply the following change of basis

R1 o
O
X

V
/

U

−1 X


R1 o

V
U

R0
O





1 p − xa 0 0


0
0
0 1

−1 X , where X = 

,
X
0
1
0 0



/
R0
0
0
1 xa

R1 = R{3 − n} ⊕ R{5 − n} ⊕ R{5 − n} ⊕ R{3 − n} and
R0 = R{6 − 2n} ⊕ R{2} ⊕ R{8 − 2n} ⊕ R.

C is of q-degree 0; a straightforward computation shows that


∗

0


−α ∗

U =
 0 ∗


0

β

0

∗




∗


β
∗

∗ α(xa − xe )(xa − xf ) ∗
49



and



∗

∗


−α −β

V =
 ∗
∗

0

0

∗

∗




0 

.
∗
∗ 

α(xa − xe )(xa − xf ) −β
0

We compute gcd(α, β) = gcd(α, (xd −xc )α−β) = gcd((xd −xa ), (xb −xe )(xb −xf )) = 1,
hence by symmetry gcd(xa − xe , β) = 1 and gcd(xa − xf , β) = 1 as well. Therefore,
gcd(α, β) = gcd(α(xa − xe )(xa − xf ), β) = 1, so we may apply Theorem 3.3.11 to get

Cp (DIIIa ){3} ∼
=
∼
=
∼
=

∗ −α
∗
∗

β

{2} ⊕

∗ α(xa − xe )(xa − xf )
∗
β

−α

∗ β + (xe + xf − xa )α
∗

xa − xd

∗ xb xc − xe xf

{2} ⊕

{2} ⊕

2
∗ α(xa − xe )(xa − xf ) − xa β
∗
β

∗ xa xb xc − xd xe xf
∗
β

It is easy to see that the first summand is isomorphic to Cp (DIIIb ) 1 {3}. Denote
the second summand by Υ{3}, so that we have Cp (DIIIa ) ∼ Cp (DIIIb ) 1 ⊕ Υ. By
=
Lemma 3.3.7, reflection along the middle strand is given by the ring homomorphism ¯ : R →
·
¯ =
R, xa = −xc , xb = −xb , xc = −xa , xd = −xf , xe = −xe and xf = −xd . Since Υ ∼ Υ by
¯
¯
¯
¯
¯
¯
Theorem 3.3.10 under this isomorphism, we obtain that Cp (DIIIa ) ∼ Cp (DIIIb ) 1 ⊕ Υ,
=
which implies claim (e).
(f) This follows immediately from (a) and (d).
We will collectively refer to diagrams of type Dr and Ds as resolved crossings.
Proof. (of Theorem 3.3.8) We will prove the theorem by repeatedly reducing Cp (D) according to Proposition 3.3.12. At each step, we get a complex of matrix factorizations whose
50

underlying graded object is

i Cp (Di ) for some collection of singular diagrams in braid

form. Following Wu [47], we define a complexity function on singular braids by i(D) =

j ij

where j runs over all resolved crossings in the diagram and ij is 1 for an oriented smoothing
and one plus the number of strands to the left of the crossing for a singular crossing. We
show that each step of the reduction process decreases either the maximum complexity of
diagrams Di or the number of diagrams that have maximum complexity. This reduction can
be performed as long as the maximum complexity is greater than 1. The only connected
diagrams of complexity 1 are Dr and Ds , so if the maximum complexity is 1, Cp (D) is
the direct sum of shifts of Cp (Dr ) and Cp (Ds ) and the Lemma follows. To perform the
reduction, choose a diagram of maximum complexity. The Lemma below guarantees that
either DIIIa or one of the diagrams on the left hand side of Proposition 3.3.12(a)-(d) or (f)
is a subdiagram of D. In the latter case we can simply replace the complex on the left hand
side by the the one on the right-hand side; notice that this reduces the number of diagrams
of this complexity. If there is a subdiagram of type DIIIa , we are given a complex of the
form
α
...C

k−1

γ δ
β
k+1
k
−−−
...,
− − → C ⊕ Cp (DIIIa ) − − − → C
−−

b
which is (up to a grading shift) isomorphic in K (hmf w (R)) to
 
α
 
β 
 
γ δ 0
1
k−1
k
k+1
...C
⊕ Cp (DIIIb ) − − → C ⊕ Cp (DIIIa ) ⊕ Cp (DIIIb ) − − − − → C
−−
−−−−
...,

51

which is in turn isomorphic to

...C

k−1

k+1
k
−
...,
⊕ Cp (DIIIb ) → C ⊕ Cp (DIIIa ) ⊕ Cp (DIIIb ) → C
−

so we once again were able to reduce the number of diagrams of the given complexity.
Lemma 3.3.13. If D is a connected singular (open) braid diagram of complexity greater
than 1, then it contains at least one of the following subdiagrams:
(i) A resolved crossing of type Dr or Ds in rightmost position which is the only resolved
crossing in this column,
(ii) a diagram Dr which has the property that D stays connected when Dr is removed,
(iii) a diagram of type DII , DIII or DIV .
Proof. We prove the lemma by induction on the braid index. If the braid index is 2 and
i(D) > 1, then we either have a subdiagram of type Dr , which can be removed without
disconnecting the diagram, or we have at least two subdiagrams of type Ds and none of type
Dr , so we can find DII as a subdiagram. If the braid index is greater than 2, we may assume
that there are at least two resolved crossings in rightmost position. We may also assume
that we have no subdiagrams of type Dr in rightmost position, since we could remove them
without disconnecting the diagram. If two such singular crossings are adjacent, we have
found DII as a subdiagram. Otherwise choose the topmost two such singular crossings and
apply the induction hypothesis to the part of the braid between those two singular crossings,
giving us either a subdiagram of the required type inside this part of the braid or, potentially
after performing an isotopy, a diagram of type DIII or DIV .
52

3.3.4

Mutation invariance of the inner tangle

Lemma 3.3.14. Let R = Q[xa , xb , xc , xd ]/(xa + xb = xc + xd ) and let ¯ be the ring
·
homomorphism defined by xa = −xb , xb = −xa , xc = −xd and xd = −xc , which induces
¯
¯
¯
¯
an involution functor on hmf 2 . Let F be the grading shift functor {2} and G be the identity
functor. Then there is a differential ∂ on the morphism spaces of hmf 2 satisfying the
hypothesis of Lemma 3.1.1

Proof. ¯ : R → R is well-defined since xa + xb = −xb − xa = −xd − xc = xc + xd . R is
·
isomorphic to the polynomial ring in xa , xb and xc . Substituting xb = −xa in any expression
of the form r − r, we obtain 0, hence r − r is divisible by xa + xb and we may define ∂ on
¯
¯
r
R by ∂r = x r−¯ . Viewing the ring R as an additive category with one element, it is
a +xb
straightforward to check that ∂ satisfies the hypothesis of Lemma 3.1.1.
The differential descends to a differential on hmf 2 . First note that objects in hmf 2 are
direct sums of one-term Koszul factorizations K(u; v) with potential w = p(xc ) + p(xd ) −
p(xa ) − p(xb ). It follows from the proof of the one-crossing case of Lemma 3.3.7 that ∂w = 0
and ∂v = 0 for the two choices of v, that is v = xc − xa and v = (xc − xa )(xc − xb ). This
implies 0 = ∂w = ∂u v + u ∂v = ∂u v, hence ∂v = 0 since R does not have zero divisors. We
¯
define the differential of a morphism of such matrix factorizations,

R{

deg v −deg u o
}
O2
y

R{

v
u

/

R
O
z

deg v−deg u
+ deg z} o
2
53

v
u
/

R{deg z}

to simply be
R{

v

deg v −deg u
+ 2} o
2O

/

u

R{2}
O

∂y
R{

∂z
v
u

deg v−deg u
+ deg z} o
2

/

R{deg z}

This is a morphism of matrix factorizations since ∂y u = ∂(yu) = ∂(u z) = u ∂z and
∂z v = ∂(zv) = ∂(v y) = v ∂y.
Since any null-homotopic morphism

R{

v

deg v −deg u o
}
O2

/

u

u h+kv
deg w
R{ 2 + deg h} o

R
O
hu+v k

v
u
/

R{deg h + deg u}

is sent to the null-homotopic morphism

R{

v

deg v −deg u
+ 2} o
2O

u ∂h+∂k v
deg w
R{ 2 + deg h} o

/

u

R{2}
O
∂h u+v ∂k

v
u
/

R{deg h + deg u}

∂ descends to a differential on hmf 2 .
The natural transformation φ is given by

deg v−deg u o
}
O2
xa +xb

v
u

deg v−deg u
+ 2} o
2

v
u

R{

R{

/

R
O
xa +xb

/

R{2}

Since we can view (representatives of) morphisms in hmf 2 as pairs of elements of R, the
54

fact that R satisfies the hypothesis of Lemma 3.1.1 implies that hmf 2 does as well.

3.3.5

Proof of invariance

Before we can finish the proof, we need to borrow another Lemma from [37].
Lemma 3.3.15. (Lemma 5.16 in [37]) Let D be the diagram of a single crossing with endpoints as in Figure 3.2(a). Then the maps xb : Cp (D){2} → Cp (D) and xc : Cp (D){2} →
Cp (D) are homotopic. Since xa + xb = xc + xd , this of course implies that xa and xd are
homotopic as well.
Proof. Let d+ : Cp (Dr ) → Cp (Ds ) be the differential of a positive crossing and d− :
Cp (Ds ) → Cp (Dr ) be the differential of a negative crossing. Clearly, d− d+ = xc − xb :
Cp (Dr ) → Cp (Dr ) and d+ d− = xc − xb : Cp (Ds ) → Cp (Ds ), so d

is a null-homotopy

for xc − xb : Cp (D± ) → Cp (D± ). We ignored q-gradings above, the reader can easily check
that the proof applies in the graded setting as well.
We are now ready to prove Theorem 3.3.1. Given a pair of mutants L1 and L2 , we may
assume, by Theorem 3.3.4, that the mutation is realized as a mutation of type Rz whose inner
b
tangle diagram D is in braid form. By Theorem 3.3.8, there is an object C in K (hmf 2 )
b
such that Cp (D) ∼ C in K (hmf w (R)). Applying the ring isomorphism ¯ we obtain an
·,
=
¯ = ¯
isomorphism Cp (D) ∼ C, hence by Lemma 3.3.7, Cp (D) ∼ C. Applying Lemma 3.1.1, we
= ¯
obtain that Cone(xa + xb : C{2} → C) is isomorphic in hmf 2 , and hence in hmf w (R) to
¯
¯
Cone(xa + xb : C{2} → C). Taking the tensor product over Q[xb , xc , xd ] with the complex
associated to the outer tangle, we get an isomorphism

◦
◦
◦
◦
Cone(xa + xb : Cp (L1 ){2} → Cp (L1 )) ∼ Cone(xa + xb : Cp (L2 ){2} → Cp (L2 ))
=
55

◦
◦
by Proposition 2.4.1, where L1 and L2 denote L1 and L2 cut open at a, respectively. Because
we consider only positive mutation, xa and xb lie on the same component of both L1 and
L2 , so xa and xb are homotopic by repeated application of Lemma 3.3.15. Hence we get an
isomorphism

◦
◦
◦
◦
Cone(2xa : Cp (L1 ){2} → Cp (L1 )) ∼ Cone(2xa : Cp (L2 ){2} → Cp (L2 ))
=

and thus
Cp (L1 )[−1]{2} ⊕ Cp (L1 ) ∼ Cp (L2 )[−1]{2} ⊕ Cp (L2 ).
=
Cp (L) is generally infinitely-generated, but since its definition involves only finitely many
generators of the form Q[x1 , . . . , xn ], where the xk all have q-degree 2, it is finite-dimensional
in every q-degree. Thus the isomorphism types of Cp (L1 ) and Cp (L2 ) are determined by
−1
−1
their Poincar´ Laurent power series χp (Lk ) ∈ Z[t , t][q , q]],
e
ij
i,j
q t dim(H (Lk )).

χp (Lk ) =
i,j
Thus the above equivalence translates to

2 −1
2 −1
(q t
+ 1)χp (L1 ) = (q t
+ 1)χp (L2 ),

which implies χp (L1 ) = χp (L2 ) and thus

Cp (L1 ) ∼ Cp (L2 ).
=

56

Proof (of Corollary 3.3.2). This follows directly from Theorem 1 in [37], which asserts that
for sufficiently large n, the sl(n) homology of a knot is a regraded version of its HOMFLY-PT
homology. It is clear that we can recover the triple grading of HOMFLY-PT homology by
choosing n large enough.

57

Chapter 4

Khovanov Homology of 3-braids

According to Garside’s solution of the word problem for Bn (see for example Birman [4],
n
Theorem 2.5), each element of Bn has a unique representative of the form ∆ s (which can
be determined algorithmically), where s is a positive braid and ∆ is the central element of
Bn , geometrically corresponding to a half twist.
From this point of view, it is natural to study how knot invariants change under insertion
and deletion of powers of ∆. In chapter, we consider the case of the Khovanov homology
of 3-braids. We show (Theorem 4.4.1) that if s is a positive 3-braid, then the Khovanov
2k
homologies of the closures of s and ∆ s are related by a distinguished triangle. If in
addition k ≥ 0, we show that the Khovanov homology of the closure of ∆

2k

s is determined

by the Khovanov homology of the closure of s.

The result is a generalization of earlier work of Turner [42], who computed the Khovanov
homology of (3, p) torus links.
58

4.1

Khovanov Homology

As before, we work in the setting of universal sl(2) homology. In contrast to the previous
treatment, we will allow arbitrary signs (+ or −) to be assigned to each crossing of the
tangle. We do not demand that these signs be induced by an orientation of the tangle.
The Khovanov complex of a single positive crossing (X+ ) is given as a chain object as
Kh(X+ ) = Kh(XA ){1} ⊕ Kh(XB )[1]{2}, where XA is the A-smoothing and XB the Bsmoothing of the crossing:

o

A

B /

The Khovanov complex of a negative crossing X− is Kh(X− ) = Kh(XA )[−1]{−2} ⊕
Kh(XB ){−1} as an chain object. Clearly Kh(X+ ) = Kh(X− )[1]{3}.
We also introduce the notation · for the Khovanov complex Kh(·) considered as an
object of the appropriate category of chain complexes (not the homotopy category). We
write the tensor product arising from gluing two braids together simply as concatenation.

4.2

Simplifying chain complexes

Since we are studying Khovanov complexes in the more general setting of complexes over a
(not necessarily abelian) additive category, we do not have the usual tools of homological
algebra at our disposal. Instead of taking homology, we will reduce complexes using a
specific type of a strong deformation retract, which allows us to state a result reminiscent of
the spectral sequence induced by a double complex.
ˆ
Definition 4.2.1. A chain map G : C → C is called a strong deformation retract if there is a
59

ˆ
ˆ
chain map F : C → C a homotopy map h : C → C[−1] such that GF = I, F G = I −dh−hd
2
and hF = 0 = Gh. If in addition h = 0 we will call G a special deformation retract (Note:
this is not a standard definition).
Remark 4.2.2. If idempotents in the underlying additive category split, then it is easy to
ˆ
see that this definition is equivalent to C being isomorphic to a direct sum of C and a complex
whose differential is the identity. The projections onto the three direct summands of C are
given by F G, dh and hd; note that dh and hd are idempotents since hdh = h(I −hd−F G) =
h.
Proposition 4.2.3. The property of being a special deformation retract is closed under
composition.
Proof. It is well-known that strong deformation retracts are closed under composition, so
2
we only need to show that h = 0. Let C1 , C2 and C3 be chain complexes. For i = 1, 2,
let Gi : Ci → Ci+1 , Fi : Ci+1 → Ci , hi : Ci → Ci+1 [−1] such that Gi Fi = I, Fi Gi =
I − di hi − hi di , hi Fi = 0 = Gi hi . Then
2
2
2
h = h1 − h1 F1 h2 G1 − F1 h2 G1 h1 + F1 h2 G1 F1 h2 G1 = F1 h2 G1 = 0.

The following theorem is essentially a homotopy version of the spectral sequence of a
double complex, but avoids the problem that it is in general not possible to reconstruct
the integral homology from the E∞ page. We adopt the following conventions. A double
complex is an object in a bigraded additive category with a horizontal differential d of
bidegree (1, 0) and a “diagonal” differential of bidegree (0, 1), in particular d
60

2

= 0 and

2
f = 0. We require that differentials anti-commute, i.e. df + f d = 0.

..

..

?.
?.












 j+1

f
/ C j+1
/ C j+1 i
/ ···
···
i
? i+1
j ?

di 







 dj
 j
 f
i+1
i / j
/ ···
/ Cj
Ci+1
···
i
?
?












.
.

..

..

The total complex is given by the direct sums

j
i+j=s Ci over the columns in the above

picture and differential d + f . Since we are not interested in the vertical and diagonal chain
complexes, we simply refer to the total complex as the double complex.

j
ˆ
Theorem 4.2.4. If C = Ci is a (bounded) double complex and Gi : Ci → Ci are special
deformation retracts with inverses Fi and associated homotopy maps hi , then (the total comˆ
plex of ) C is homotopy equivalent to C, which is given by

j
i+j=s Ci and has differential

ˆ
d + G(f + f hf + . . . )F In fact, this homotopy equivalence is a special deformation retract in
the sense of Definition 4.2.1.

2
I
Proof. It is convenient to formally define I−x = I + x + x + . . . . We first need to show
ˆ
ˆ
that the map d + Gf I F = d + G I f F does indeed define a differential, i.e. that
I−hf
I−f h
2
ˆ
ˆ
its square is 0. Clearly, f d = −df , Gd = dG, F d = dF , F G = I + dh + hd and f = 0.
61

Therefore

ˆ
d+G
= G df

I fF
I−f h

ˆ
d + Gf

I
I−hf

I + I f d + I f (I + dh + hd)f I
F
I−hf
I−f h
I−f h
I−hf

= G

I ((I − f h)df + f d(I − hf ) + f (dh + hd)f ) I F
I−f h
I−hf

= G

I (df − f hdf + f d − f dhf + f dhf + f hdf ) I F = 0
I−f h
I−hf

We will show that the following picture defines a homotopy equivalence.

Co
O
I F
I−hf

G

d+f
/

I
I−f h

O

I F
I−hf



ˆ
C

C[1]

−h−hf h−...

ˆ
d+G(f +f hf +... )F

/

G

I
I−f h



ˆ
C[1]

The upward-pointing arrows define a chain map since

(d + f )

I F − I F d + Gf I F
ˆ
I−hf
I−hf
I−hf

=

I ((I − hf )(d + f ) − d(I − hf ) − F Gf ) I F
I−hf
I−hf

=

2
I
I F =0
d + f − hf d − hf − d + dhf − f − dhf − hdf
I−hf
I−hf

Similarly, we get a chain map from the downward-facing arrows because

G
= G

I (d + f ) − d + G I f F G I
ˆ
I−f h
I−f h
I−f h
I ((d + f )(I − f h) − (I − f h)d − f F G) I
I−f h
I−f h
62

= G

2
I
I =0
d + f − df h − f h − d + f hd − f − f dh − f hd
I−f h
I−f h

Finally, we need to show that

I F G I − I − (d + f )h I − I h(d + f )
I−hf
I−f h
I−f h I−hf
=

I
F G − (I − hf )(I − f h)
I−hf
−(I − hf )(d + f )h − h(d + f )(I − f h)

=

I
I−f h

2
I
I + dh + hd − I + hf + f h − hf h
I−hf
2
2
−dh − f h + hf dh + hf h − hd − hf + hdf h + hf h

4.3

I =0
I−f h

Three-Braids

We are interested in the effect of adding a number of twists to the Khovanov homology (or
more precisely, the homotopy type of the Khovanov complex) of the closure of a 3-braid.
One can view braids as links in a standardly embedded thickened annulus. Adding twists
then corresponds to switching to a non-standard embedding of the annulus. Any smoothing
of a closed 3-braid in an annulus that is not the trivial 3-braid yields isotopic links regardless
of the chosen embedding. This fact can be exploited to give a recursive formula for the
√ w(s)
6
2
2
6
Jones polynomial, namely J(∆ s) − t J(s) = (− t)
(J(◦ ◦ ◦) − t J(∆ )). The goal of
this chapter is to establish a similar relationship for Khovanov homology. We will restrict
ourselves to positive 3-braids.
2
3
In the following we will always represent a full positive twist ∆ by the braid (σ1 σ2 )
63

and a full negative twist ∆

−2

−3
−1 −1 3
by the braid (σ1 σ2 )
= (σ2 σ1 ) .

−2
∆

2
∆

If s is a positive 3-braids of length n, we will compare s with ∆

2k

s (k ∈ Z). In light of the

k
2k
2k
previous discussion, C := Kh(∆ s) can be viewed as a tensor product of Kh(∆ ) and
k
2k
Kh(s), which is a double complex whose diagonals are CX := Kh(∆ sX )[b(X)]{2b(X)+n}
n
with differential d, where sX is the smoothing corresponding to X ∈ {A, B} and b(X) is
the number of Bs occurring in X. Horizontal maps between smoothings X and X are
k
k
, which is induced by a crossing change on s. Let C ¯n be the subcomplex
A
X,X
k
k
k
of C whose underlying graded object is
n
n C and CA be the quotient
X∈{A,B} \{A } X
k
k
k k
complex C /C ¯n , which we identify with C n . We may think of C ¯n as a cube of partial
A
A
A
k
2k
n
resolutions for C = Kh(∆ s) with the A -vertex removed.
given by f

2
Figure 4.1: Transforming ∆ sX into sX . All other cases, negative twist, first B-smoothing
on the bottom are analogous.

In order to apply Theorem 4.2.4, we need to establish special deformation retractions
k
n
k
˜k
from all the CX . If X = A , then CA retracts to a complex CA , which is (for positive
k) supported in gradings between 0 and 4k by Corollary 4.4.2 below. If X is not the allA smoothing, then the braid ∆

2k

sX is isotopic to sX , with an explicit isotopy given by

Figure 4.1. Note that the isotopy consists of only Reidemeister I and II moves, which induce
64

/

⊕ O ⊕

( −h

−h
−
/

⊕



⊕

0
0

− )
/

−

(0

0)

/

Figure 4.2: Reidemeister II moves induce special deformation retracts
special deformation retracts. This is obvious for Reidemeister I moves, since homotopies are
of homological degree −1 and the complex is supported in two adjacent homological heights.
For Reidemeister II, the top row of Figure 4.2 shows the complex
which is isomorphic to a direct sum of

after delooping,

and a complex with identity differential, as seen

in the figure. We leave it to the reader to work out grading shifts. In order to be able to
perform this isotopy, we need to change 4k positive crossings to negative crossings, thus we
k
˜k
˜k
get a retraction G : CX → CX . where CX := Kh(sX )[b(X) + 4k]{2b(X) + n + 12k}. Note
0
˜k
that CX = CX [4k]{12k}.
˜k
Theorem 4.2.4 now implies that there is a reduced complex C , which is the direct sum
˜k
of the CX s with differential d +

k
k
˜k
¯
k G(f h) f F . As for C , we define a subcomplex CA and

˜k
˜k k
a quotient complex CA = C /CA .
¯
m
Proposition 4.3.1. Let m > 1. Then, in the notation of Theorem 4.2.4, G(f h) f F = 0
m
˜k
if k ≥ 0. If k < 0, G(f h) f F = 0 only on CA .
¯
m
Proof. The chain morphism G(f h) f F has (homological) degree 1 since it is a differential.
n
˜k
Since all the CX (X = A ) are supported in grading b(X) + 4k, a degree-1 morphism that
65

˜k
travels along more than one edge of the cube is 0 if it starts at any place other then CA . If
˜k
˜k
k ≥ 0, then CA is supported in gradings ≤ 4k, so the morphisms originating at CA are also
0 if m > 1.
k
k
0
k
Proposition 4.3.2. Restricted to CA , Gf
F = ±f
[4k]{12k}, where f
:
¯
X,X
X,X
X,X
k
k
k
CX → C
is an edge belonging to the differential of C as above.
X
Proof. This is trivial if there is a B-smoothing to the left of the crossing that is being
2k
changed, since it does not affect the sequence of Reidemeister moves performed on ∆ .
If changing the smoothing changes the left-most B-smoothing from being on the bottom
k
F is induced by a morphism
→
of q-degree 1.
X,X
k
F is an integer multiple of the
The space of such morphisms is one-dimensional, so Gf
X,X
k
F = mg. As a special case of
morphism induced by the saddle g =
→
, say Gf
X,X
k
F is homotopy equivalent to the cone
Theorem 4.2.4, the cone over the closure of Gf
X,X
k
, which computes the closure of
over f
, that is an unknot. If m were
X,X
k
F = 0, so the homology of the
neither 1 nor −1, then with coefficients in Z/mZ, Gf
X,X

to being on the top, Gf

(mod-m) Khovanov complex of the unknot would have rank 4 + 2 = 6, a contradiction. The
k
F :
case where Gf
X,X

→

is completely analogous.

If the left-most B-smoothing stays on the top or on the bottom, say without loss of
k
generality on the top, it is easy to calculate Gf
F :
→
X,X
k
2k
2k
that by neck-cutting, the morphism f
: ∆
→ ∆
X,X

explicitly. Notice
can be written as

I 2k f1 + I 2k f2 − h I 2k f3 , where f1 =
, f2 =
and f3 =
(the middle
∆
∆
∆
parts of these cobordisms correspond to ,
and , respectively). Clearly, I 2k f2 ◦ F =
∆
F ◦ f2 and I 2k f3 ◦ F = F ◦ f3 . Since sliding a dot across a crossing gives a chain
∆
morphism that is homotopic to the negative of the original one, it is easy to see that I 2k f1 ◦
∆
66

k
F ◦ f1 . Thus Gf
F = G ◦ I 2k f1 ◦ F + I 2k f2 ◦ F − h I 2k f3 ◦ F
G◦
X,X
∆
∆
∆
0
0
F ◦ f1 + F ◦ f2 − h F ◦ f3 = GF f
= ±f
.
and
are supported
X,X
X,X

F

in a single homological grading, hence the notions of homotopy and equality coincide and
Gf

k
0
F = ±f
.
X,X
X,X

Proposition 4.3.3. The following isomorphisms of complexes hold:
˜k = 0
(a) CA ∼ CA [4k]{12k}.
¯
¯
˜k
(b) If k > 0, then C is isomorphic to the mapping cone

0
˜k
Cone f : CA → CA [4k]{12k} ,
¯
n
k
˜k = 0
˜k
where f is induced by GfA,X F : CA → CX ∼ CX [4k]{12k} for all X ∈ {A, B}
with b(X) = 1.
Proof. We prove both parts in parallel. In each case, the two complexes agree up to signs.
˜k
More precisely, the complexes on the left are given as cubes with vertices CX (and with
n
k
F . By
the A vertex removed in case (a)), whose edges are the chain morphisms Gf
X,X
k
F
Proposition 4.3.1, the corresponding morphisms on the right hand side are
Gf
X,X
X,X
∗
with
∈ Z = {±1}. We claim that when viewing this cube (possibly with a vertex
X,X
and all its adjacent cells removed) as a simplicial complex, defines a 1-cocycle. This requires
˜
us to show that if a face of the cube has vertices X, X , X and X

that

=
X,X X ,X
k
k
˜ X ,X , which will follow once we’ve shown that GfX ,X F ◦ GfX,X F = 0. If
˜
X,X
n
0
0
X = A , then this morphism is a shift of ±f
f
, which cannot be zero as a
X ,X X,X
n
composition of two saddle cobordisms. If X = A , we will argue by contradiction, so
k
k
k
k
assume that Gf
F ◦ Gf n
F = Gf
f n
F = 0. Extend s by σ1 on the
X ,X
A ,X
X ,X A ,X
67

k
k
right and note that the above equality implies that Gf
f n
F 0. Again,
X B,X B A B,X B
0
0
0
0
up to grading shift this morphism is given by f
f n
:C n →C
.
A B
X B,X B A B,X B
X B
0
As before, this morphism is not zero, and it cannot be homotopic to zero either, since C n
A B
0
and C
are supported in a single homological grading, which is the desired contradiction.
X B
Since the simplicial complex is contractible in both cases and thus has trivial first cohomology,
X,X

4.4

is a coboundary and there exists a 0-cochain η such that ∂η =

= ηX η

X

and thus

. Hence η gives the desired isomorphism of complexes.

Khovanov homology after adding twists

2k
We are now ready to prove the promised relationship between Kh(s) and Kh(∆ s).

Theorem 4.4.1. Let ∆ = σ1 σ2 σ1 ∈ B3 be a half-twist and let s denote the closure of the
3-braid s. Suppose that all three-braids are oriented in the natural way and define



r(s) = σi


σ1 σ2

if s =
if s only contains
if s contains both

σi ’s
σ1 ’s and σ2 ’s

(a) There is a chain complex Ys such that

Ys ⊕ Kh(r(s)){|s| − |r(s)|} ∼ Kh(s)
=

(b) For any k ∈ Z, there is a distinguished triangle

2k
2k
Ys [4k]{12k} −→ Kh(∆ s) −→ Kh(∆ r(s)){|s| − |r(s)|} −→ Ys [4k + 1]{12k}.
−
−
−
68

ÔB
ÔÔ
ÔÔ =
ÔÔ
f1 ÔÔÔÔ
ÔÔ
ÔÔ
qq8
ÔÔ qqqq
Ô
q
ÔÔ qqq
ÔÔ qqq f1
q
XXwww
XX wwwwf2
XX www
www
XX
w&
XX
XX
X
f2 XXX
XX !
XX
XX


.
.
.

.
.
.

………… g
…………
1…
………*
i4
iiii
iiii
iii −g
1
xxx
xxx
xxx
x
h1 xxxxx'
p7
h2 ppppp
p
p
ppp
ppp
p
………… g
…………
2…
………*
iii4
iiii
iiii −g
i

2

.
.
.

.
.
.
.
.
.
.
.
.
.
.
.

.
.
.

Figure 4.3: The Khovanov complex near a corner of the Khovanov cube
(c) If k ≥ 0, then

2k
2k
Kh(∆ s) ∼ Ys [4k]{12k} ⊕ Kh(∆ r(s)){|s| − |r(s)|}
=

Proof. Introduce morphisms fi , gi and hi as shown in Figure 4.3. The diagrams represent
smoothings of the closure of the braid even though they are depicted as smoothings of braid
0
diagrams. We will decompose C := C into A ⊕ B as shown in Figure 4.4.
Now, z ∈ C0 =

implies

dz = ((f1 z, . . . f1 z), (f2 z, . . . f2 z)) ∈ A1 .

Similarly,

d((x, . . . x), (y, . . . y)) = ((g1 x − g1 x, . . .), (h1 x − h2 y, . . .), (g2 y − g2 y, . . .)) ∈ A2 .
69

A0 = C0
A1 = ((x, . . . x), (y, . . . y)) ∈
A2 =

m

⊕

((0, . . . 0), (y, . . . y), (0, . . . 0)) ∈

Ai = 0
B0 = 0

n
m
2 ⊕

⊕

n
2

(i > 2)

B1 = ((x1 , . . . xm−1 , 0), (y1 , . . . ym−1 , 0)) ∈
B2 =

mn

m

⊕

n

x1 , . . . x m , (y1 , . . . ymn−1 , 0), z1 , . . . z n
2
2
m
n
mn
2 ⊕
2
∈
⊕

Bi = Ci

(i > 2)

Figure 4.4: Decomposing the Khovanov complex of a closed braids into a direct sum
2
We claim that d|A : A1 → A2 is surjective, which implies dA2 ⊆ A3 = 0 since d = 0. It
1
is clearly sufficient to consider the case n = m = 1, where we consider the homology of the
Khovanov complex of an unknot. Since A1 = C1 for n = m = 1 and the second homology of
the complex associated to the unknot is trivial, im(d|A ) = ker(d|A ) = A2 , which implies
1
2
the claim. Hence dA ⊆ A.
If z = ((x1 , . . . xm ), (y1 , . . . ym )) ∈ B1 , then dz = ((. . .), (. . . , h1 0 − h2 0), (. . .)) ∈ B2 .
Thus dB ⊆ B. We can now set Ys := B to prove the claim.
˜
˜
˜k
For part (b), notice that we can find a similar subcomplex B in C := C . By Propo˜ =
˜ ˜ = ˜
˜
sition 4.3.3, B ∼ B[4k]{12k}, furthermore C/B ∼ A{|s| − |r(s)|}, where A is the reduced complex corresponding to ∆

2k

r(s). Thus the short exact sequence of complexes

˜
˜
0 → B[4k]{12k} → C → A{|s| − |r(s)|} → 0 gives rise to the desired distinguished triangle.
˜
For part (c), we can construct A in the same way as A in part (a), and we get a similar
˜
˜
decomposition C = A{|s| − |r(s)|} ⊕ B[4k]{12k} that carries over to homology.
70

[0]{0}
‡

‡‡‡‡‡
‡‡‡‡‡
‡‡‡‡‡
‡‡‡‡

[0]{1}

‡‡‡‡‡
‡‡‡‡‡
‡‡‡‡‡
‡‡‡‡‡

[0]{2}

[1]{2} =

[4]{11}

[1]{3} =

[3]{9}

2
Figure 4.5: (Partial) tree of resolutions of ∆

2k
Corollary 4.4.2. For k ≥ 1, the homotopy type of the Khovanov complex of ∆ is supported
in homological gradings between 0 and 4k.
Proof. For k = 1, the proof follows from Figure 4.5 and the fact that a tangle with n positive
crossings is supported in gradings between 0 and n by repeated application of Theorem 4.2.4.
4
For the induction step, notice that (σ1 σ2 ) is supported in gradings between 0 and 8 ≤ 4k+4,
2k
so for s = ∆ , Ys is supported between 0 and 4k by induction hypothesis, so Ys lies between
2k
2k+1
4 and 4k+4 and ∆ r(s) lies between 0 and 4, thus by Theorem 4.4.1(c), ∆
is supported
in gradings between 0 and 4k + 4.
Remark 4.4.3. We are using Corollary 4.4.2 to show Theorem 4.4.1, which might appear
to be circular reasoning. However, since we only use Theorem 4.4.1 for k − 1 to show
Corollary 4.4.2 for k, we can use induction to show Theorem 4.4.1 and Corollary 4.4.2
simultaneously.

71

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72

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