A KIRBY COLOR FOR KHOVANOV HOMOLOGY MATTHEW HOGANCAMP DAVID E. V. ROSE AND PAUL WEDRICH Abstract. We construct a Kirby color in the setting of Khovanov homology an ind-object of the

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A KIRBY COLOR FOR KHOVANOV HOMOLOGY

MATTHEW HOGANCAMP, DAVID E. V. ROSE, AND PAUL WEDRICH

Abstract. We construct a Kirby color in the setting of Khovanov homology: an ind-object of the

annular Bar-Natan category that is equipped with a natural handle slide isomorphism. Using functo-

riality and cabling properties of Khovanov homology, we deﬁne a Kirby-colored Khovanov homology

that is invariant under the handle slide Kirby move, up to isomorphism. Via the Manolescu–Neithalath

2-handle formula, Kirby-colored Khovanov homology agrees with the gl2skein lasagna module, hence

is an invariant of 4-dimensional 2-handlebodies.

Contents

1. Introduction 1

2. Categorical Background 5

3. Dotted Temperley–Lieb and annular Bar-Natan categories 12

4. Kirby color and handle slides 22

5. Diagrammatic presentation 28

6. Kirby-colored Khovanov homology 40

7. Future directions 43

References 45

1. Introduction

In the context of quantum link invariants (such as the Jones polynomial) a Kirby color1is a certain

linear combination of cabling patterns yielding a framed link invariant that is invariant under the

second (handle slide) Kirby move [Kir78]. The resulting “Kirby-colored” quantum invariant of a link

L ⊂ S3can then be regarded as an invariant of the 4-manifold obtained by attaching 2-handles to the

4-ball B4along the link L ⊂ ∂B4[Lic92]. Perhaps the most famous examples of this process are the

Witten–Reshetikhin–Turaev invariants [Wit89,RT91], though we note these are special in that they

admit a further renormalization that is invariant under the ﬁrst (blow-up) Kirby move2.

Khovanov homology [Kho00] is a bigraded homology theory for links L ⊂ S3, which categoriﬁes

the Jones polynomial. One of the longest-standing problems regarding Khovanov homology is whether

(and if so, how) it extends to an invariant of 3- or 4-manifolds. The goal of this paper is to propose a

solution to this problem that proceeds by developing a Kirby color for Khovanov homology.

1.1. The Kirby color. To describe our work with slightly more precision, recall that in the (pre-

categoriﬁed) context of framed link invariants associated with a linear ribbon category, the set of cabling

patterns forms an algebra. For the Jones polynomial, this is the Kauﬀman bracket skein algebra of

the thickened annulus. Analogously, in the categoriﬁed context the cabling patterns form a monoidal

1This name seems to have come into use sometime before 2001 [Bla03], although the concept is older. Another

common name in the tensor category literature is “(virtual) regular object”, see e.g. [EGNO15, p.270]

2Being invariant under both Kirby moves, the Witten–Reshetikhin–Turaev invariants are thus 3-manifold invariants,

depending only on the boundary of the aforementioned 4-dimensional 2-handlebody.

arXiv:2210.05640v2 [math.GT] 25 Oct 2022

2 MATTHEW HOGANCAMP, DAVID E. V. ROSE, AND PAUL WEDRICH

category: for Khovanov homology this role is played3by the Bar-Natan category of the thickened

annulus, which hereafter will be denoted ABN . For this introduction, it suﬃces to note that objects

of ABN are embedded 1-manifolds in the annulus and morphisms are formal linear combinations of

(dotted) cobordisms embedded in the thickened annulus, modulo local relations (from [BN05,§11.2]).

Moreover, ABN is Z-graded and linear, with monoidal structure given by inserting one annulus into

the interior of another. See §3.1 for the precise deﬁnitions.

We let cdenote the object of ABN which is a single essential circle, so cndenotes nconcentric

essential circles. The symmetric group Snacts on cn, thus the symmetric power Symn(c) exists in the

Karoubi completion Kar(ABN ) as the image of the symmetrizing idempotent.

Deﬁnition 1.1. Let ω:= ω0⊕ω1, where the summands are the colimits

ω0:= colim Sym0(c)→q−2Sym2(c)→q−4Sym4(c)→ · · · 

ω1:= colim q−1Sym1(c)→q−3Sym3(c)→q−5Sym5(c)→ · · · ,

regarded as objects of an appropriate completion ABN (see Convention 4.1). Here, the maps are given

by dotted annulus cobordisms, and the grading shifts q−nensure that these maps are degree zero.

The object ω∈ ABN is the titular Kirby color for Khovanov homology. In contrast to the pre-

categoriﬁed situation, it is an object of a monoidal category rather than an element of an algebra, and

it manifestly does not require working at a root of unity. In particular, the Kirby color ωfor Khovanov

homology does not categorify any Kirby element for the sl2Witten–Reshetikhin–Turaev invariants!

Nevertheless, the Kirby color ωbehaves like a categoriﬁed Kirby element [Vir06] in the sense that

it leads to link invariants that are invariant under handle slide. In the categorical setting, handle slide

invariance is an additional structure, rather than a property. This structure is best phrased in the

annular setting, namely using the relative Bar-Natan category of the annulus PBN , wherein objects

are annular curves with boundary (see §4.2). This category is a module category for ABN and contains

two special objects

L:= •, R := •.

Theorem A (Handle slide).The Kirby color ωis equipped with the following handle slide structure:

•(Lemma 4.13) There exists a distinguished isomorphism ω•L∼

=ω•Rin an appropriate

completion PBN , that we call the elementary handle slide. Graphically, this may be depicted

as in the ﬁrst isomorphism of (1).

(1) •ω∼

=•ω , •ω

D∼

=•

ωD

•(Lemma 4.14) Compositions of elementary handle slides assemble into a collection of handle

slide isomorphisms, which are natural with respect to cobordisms involving the “sliding strands”

(illustrated as Din the second isomorphism of (1)).

In Theorem 4.15, we further show that the Kirby color ω, together with its handle slide isomorphisms,

constitutes an object of the Drinfeld center of the completed relative Bar-Natan category PBN of

3See Remark 6.6 for a more sophisticated choice, which will not be necessary for the purpose of this paper.

A KIRBY COLOR FOR KHOVANOV HOMOLOGY 3

the annulus, considered as a bimodule for the monoidal Bar-Natan bicategory BN associated to the

rectangle. The proof of Theorem Auses a generators and relations presentation of a certain subcategory

of PBN that we provide in Theorem 4.12.

1.2. Manifold invariants and TQFT context. In order to port our results on handle slide invari-

ance from the annular setting to links in S3, we need a well-deﬁned notion of cabling in Khovanov

homology. The following is a straightforward consequence of the functoriality of Khovanov homology:

Theorem B (Cabling in Khovanov homology).Let L=K1∪ · · · ∪ Krbe an r-component framed

oriented link in S3. There is a functor

KhL:ABN ×r→VectZ×Z

sending (ck1,· · · , ckr) to the Khovanov homology of the cable Lk:= Kk1

1∪ · · · ∪ Kkr

Here, Kki

idenotes the ki-fold parallel cable of the component Kiand VectZ×Zdenotes the category

of bigraded vector spaces. The functor in Theorem Brequires certain choices that ﬁx the (original)

well-known sign ambiguity in the functoriality of Khovanov homology; this is discussed further in §6.1

where we re-state and prove this result as Theorem 6.1. Since VectZ×Zis closed under all of the relevant

operations used to deﬁne ABN (grading shifts, direct sums and summands, ﬁltered colimits), cabling

extends to a functor

KhL: (ABN )×r→VectZ×Z.

Deﬁnition 1.2 (Kirby-colored Khovanov homology).Let Lbe a framed oriented link in S3with a

decomposition into sublinks L=L1∪ L2. Set

Kh(L1∪ Lω

2) := KhL1∪L2(c, . . . , c, ω, . . . , ω)

in which all the components of L1carry the label c, and all the components of L2carry the label ω.

Remark 1.3. In [GLW18,§7.5] Grigsby–Licata–Wehrli consider a diﬀerent colimit of Khovanov ho-

mologies of cables of a knot K, which appears to be unrelated to our Kirby colored Khovanov homology.

Theorems Aand Bsuggest that Kh(L1∪ Lω

2) may be a diﬀeomorphism invariant of the pair

(B4(L2),L1), where B4(L2) is the 4-dimensional 2-handlebody obtained by attaching 2-handles to

B4along the components of L2, and L1is regarded as a link in the boundary 3-manifold S3(L2) :=

∂B4(L2). To establish this directly, one would further need to incorporate 1- and 3-handles in a way

that they satisfy an appropriate cancellation property with ω.

The resulting 4-manifold invariant would a priori be valued in isomorphism classes of bigraded vector

spaces, rather than valued in bigraded vector spaces. An upgrade to the latter would in particular

require not only isomorphisms associated to handle slide Kirby moves, but a coherent family thereof.

For this, one would need a classiﬁcation of movie moves for Kirby moves, in analogy with the Carter–

Saito movie moves for isotopic link cobordisms [CS98]. We are not aware of such a classiﬁcation.

In the absence of both of the above, we instead establish Kh(L1∪ Lω

2) as a bigraded vector space-

valued invariant of (B4(L2),L1) by comparison with the glNskein lasagna module 4-manifold invariants

from [MWW19], whose invariance is manifest.

Theorem C. Let L=L1∪ L2be a framed oriented link in S3. Decorate all the components of L1

with c, and all the components of L2with ω. The following bigraded vector spaces are isomorphic:

(1) the Kirby-colored Khovanov homology Kh(L1∪ Lω

2),

(2) the cabled N= 2 Khovanov–Rozansky homology of L1∪Lω

2as deﬁned by Manolescu–Neithalath

[MN22] for split unions L1t Lω

2and extended to the general case in [MWW22], and

(3) the N= 2 skein lasagna module (degree zero blob homology) of (B4(L2),L1) from [MWW19].

As a consequence, the Kirby-colored Khovanov homology Kh(L1∪ Lω

2) is an invariant of the pair

(B4(L2),L1), valued in bigraded vector spaces.

4 MATTHEW HOGANCAMP, DAVID E. V. ROSE, AND PAUL WEDRICH

The isomorphisms (2) ∼

=(3) have been established in [MN22,MWW22]. In §6, we verify (1) ∼

=(2) by

showing that cabling with the Kirby color ωimplements the Manolescu–Neithalath 2-handle formula

from [MN22] that deﬁnes (2).

Remark 1.4. The sl2-version of the skein lasagna module of (B4(L2),L1) is graded by the relative

second homology group H2(B4(L2),L1;Z/2). The isomorphism from Theorem Cidentiﬁes this grading

with the direct sum decomposition of KhL1∪L2(c, . . . , c, ω, . . . , ω) inherited from ω=ω0⊕ω1.

1.3. Diagrammatics. A theorem of Russell [Rus09] implies that the monoidal category ABN admits

a diagrammatic presentation in terms of the dotted Temperley-Lieb category dTL (see our §3.1 for

a deﬁnition). The latter is a non-semisimple Z-graded monoidal category that contains the familiar

Temperley-Lieb category as its degree zero subcategory. A further goal in the present paper is to

extend the graphical calculus for dTL to give a presentation of the monoidal category obtained from

ABN by adjoining the Kirby objects ω0and ω1. This is accomplished in §5.

One interesting feature of this extended calculus is the necessity of inﬁnite sums of diagrams. The

following gives the ﬂavor of the sort of relations one encounters:

[0]

n≥0

(−1)(n

[0]

•

Here, the [0]-label indicates the Kirby object ω0, and this equation gives a decomposition of the

identity morphism of ω0⊗ω0into mutually orthogonal idempotents. This relation reﬂects a certain

quasi-idempotence property of the Kirby object, namely that ω0⊗ω0∼

=Ln≥0q−2nω0(with similar

statements for ωi⊗ωj). See Corollary 5.25 for details.

1.4. The Kirby color as representing planar evaluation. For the remainder of this introduction,

we phrase all of our results in terms of the diagrammatic category dTL, instead of ABN (to which it

is equivalent). If Uis the 0-framed unknot, the associated the cabling functor KhU: dTL →VectZ×Z

satisﬁes

KhU(cn)∼

=q−nK[x1, . . . , xn]/(x2

i= 0) .

We refer to KhUas the polynomial representation of dTL (or its completion dTL) and we denote it

by Pol(−) = KhU(−). Since the cohomological grading of KhU(X) is trivial for all X∈dTL, we

typically regard Pol(−) as taking values in the category VectZof singly-graded vector spaces. The

category-theoretic role of the Kirby color ωis that it represents the functor

Pol∗: dTLop →VectZ, X 7→ Pol(X)∗

where on the right-hand side Pol(X)∗denotes the graded dual of the graded vector space Pol(X).

Speciﬁcally, in §5.3 we restate and prove the following.

Theorem D. There is an isomorphism HomdTL(−, ω)∼

=Pol∗(−) of functors dTLop →VectZ. Addi-

tionally, we have isomorphisms

ω⊗X∼

=Pol(X)⊗ω

natural in X∈dTL.

Remark 1.5. We would like to emphasize that the surprising feature in representing the functor Pol∗

is not that a representing object exists in some appropriate completion of dTL. Rather, it is that the

representing object ωcan be described very explicitly as the, arguably, simplest non-trivial directed

system (i.e. ind-object) over dTL. Furthermore, the explicit description is essential for computations

of Kirby-colored Khovanov homology (Deﬁnition 1.2) and for the diagrammatics described in §1.3.

A KIRBY COLOR FOR KHOVANOV HOMOLOGY 5

The functor Pol: dTL →VectZalso has an algebraic description via sl2representation theory. As

mentioned above, the usual (undotted) Temperley-Lieb category TL (at circle-value 2) can be regarded

as the subcategory of degree zero morphisms in dTL. As is well-known, there is a fully faithful monoidal

functor TL →Rep(sl2) that sends c7→ V, the deﬁning 2-dimensional representation of sl2. If we forget

the action of the Chevalley generators E, F ∈sl2but remember the weight grading, we obtain a functor

TL →VectZthat coincides with Pol|TL. Now Pol: dTL →VectZmay be thought of as the extension

of this functor to dTL, deﬁned by sending the “dot” endomorphism of cto the action of E∈sl2on V.

Remark 1.6. Since the Karoubi (idempotent) completion of TL is semisimple, it is not hard to see

that the restricted functor Pol∗|TL : TLop →VectZis representable by the object

n≥0

PolSymn(c)∗⊗Symn(c)

where here Symn(c) is the object in Kar(TL) corresponding to the simple ﬁnite-dimensional sl2-module

Symn(V). Since PolSymn(c)has graded dimension equal to the quantum integer [n+1], this formula

is reminiscent of the familiar formula for the Kirby element in sl2Witten–Reshetikhin–Turaev theory.

However, it is the category dTL (non-semisimple even after Karoubi completing) that naturally

arises in Khovanov homology, and the representing object for Pol∗: dTL →VectZneed not be (and

indeed is not) isomorphic to the representing object for its restriction to TL. This elucidates why our

Kirby color does not categorify the familiar Kirby element from sl2Witten–Reshetikhin–Turaev theory.

Conventions. All results in this paper hold over any ﬁeld Kof characteristic zero. We let denote

an arbitrary ﬁeld. Knots and links are always framed and oriented.

Acknowledgements. We thank Christian Blanchet, Elijah Bodish, Ben Elias, Eugene Gorsky, Jiuzu

Hong, David Reutter, Kevin Walker, and Hans Wenzl for useful discussions and correspondence. Special

thanks to Ciprian Manolescu and Ikshu Neithalath, whose 2-handle formula motivated the construction

of the Kirby color.

Funding. D.R. was partially supported by NSF CAREER grant DMS-2144463 and Simons Collab-

oration Grant 523992. P.W. acknowledges support by the Deutsche Forschungsgemeinschaft (DFG,

German Research Foundation) under Germany’s Excellence Strategy - EXC 2121 “Quantum Universe”

- 390833306.

2. Categorical Background

In this section we recall category-theoretic constructions that will be used throughout. In particular,

we review graded, linear categories and discuss (co)limits and various completions in this setting.

2.1. Graded linear categories. We begin with some basic notions, mostly for the purpose of estab-

lishing notation. Let Γ be an abelian group and a ﬁeld. A Γ-graded vector space is a Γ-indexed

collection of -vector spaces (Vi)i∈Γ. Given a pair of Γ-graded vector spaces V, W , we let HOM(V, W )

denote the Γ-graded vector space which is given in degree i∈Γ by

HOMi(V, W ) := Y

j∈Γ

Hom (Vj, Wi+j).

We let VectΓ

0denote the category with objects Γ-graded vector spaces and with morphisms

HomVectΓ

0(V, W ) := HOM0(V, W ).

The category VectΓ

0is symmetric monoidal, with tensor product given by

(Vi)i∈Γ⊗(Wj)j∈Γk:= M

i+j=k

Vi⊗Wj.

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AKIRBYCOLORFORKHOVANOVHOMOLOGYMATTHEWHOGANCAMP,DAVIDE.V.ROSE,ANDPAULWEDRICHAbstract.WeconstructaKirbycolorinthesettingofKhovanovhomology:anind-objectoftheannularBar-Natancategorythatisequippedwithanaturalhandleslideisomorphism.Usingfuncto-rialityandcablingpropertiesofKhovanovhomology,wedeneaKirby-c...

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A KIRBY COLOR FOR KHOVANOV HOMOLOGY MATTHEW HOGANCAMP DAVID E. V. ROSE AND PAUL WEDRICH Abstract. We construct a Kirby color in the setting of Khovanov homology an ind-object of the

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