Khovanov homology and cobordisms between split links
KKHOVANOV HOMOLOGY AND COBORDISMS BETWEEN SPLIT LINKS
ONKAR SINGH GUJRAL AND ADAM SIMON LEVINE
Abstract.
In this paper, we study the (in)sensitivity of the Khovanov functor to four-dimensionallinking of surfaces. We prove that if L and L (cid:48) are split links, and C is a cobordism between L and L (cid:48) that is the union of disjoint (but possibly linked) cobordisms between the components of L andthe components of L (cid:48) , then the map on Khovanov homology induced by C is completely determinedby the maps induced by the individual components of C and does not detect the linking betweenthe components. As a corollary, we prove that a strongly homotopy-ribbon concordance (i.e., aconcordance whose complement can be built with only 1- and 2-handles) induces an injection onKhovanov homology, which generalizes a result of the second author and Zemke. Additionally, weshow that a non-split link cannot be ribbon concordant to a split link. Introduction
Khovanov homology is a link homology theory, discovered by Mikhail Khovanov [Kho00], that isfunctorial under link cobordisms. It associates to any link L ⊂ R a bigraded group called Kh( L )(and, more generally, for any commutative ring R , a bigraded R -module Kh( L ; R )). Further, asmoothly embedded cobordism C in R × [0 ,
1] between links L and L (i.e., a smoothly embedded,oriented surface in R × [0 ,
1] with oriented boundary − L × { } ∪ L × { } ) induces a map Kh( C ) :Kh( L ; R ) → Kh( L ; R ) that is an invariant of C under smooth isotopies rel boundary. In this paper,we shall prove that the Khovanov functor is insensitive to the 4-dimensional linking of cobordismsbetween split links, in a sense that we now make precise. Definition 1.1. A splitting of a link L is a decomposition L = L ∪ · · · ∪ L k , where L , . . . , L k arelinks, such that L , . . . , L k are contained in disjoint 3-balls B , . . . , B k ⊂ R . L , . . . , L k are calledthe parts of the splitting. A total splitting is a splitting in which each of the parts is a knot.Given links L and L with partitions L = L ∪ · · · ∪ L k and L = L ∪ · · · ∪ L k (which need notbe splittings), a cobordism C ⊂ R × [0 ,
1] is called partition-preserving if it is a disjoint union ofsurfaces C , . . . , C k (the parts of C ), where C i is a cobordism from L i to L i . The cobordisms C i are not required to be connected. We say the cobordism is split if L and L are split links (usingthe same family of 3-balls B , . . . , B k ) and each C i is contained in B i × [0 , C = C ∪ · · · ∪ C k and D = D ∪ · · · ∪ D k from L to L are called partition-homotopic if there is a smooth homotopy from C to D (rel boundary) that is anisotopy when restricted to each of the components C i . This is equivalent to the statement that foreach i , C i is isotopic rel boundary to D i .The following is the main theorem of the paper. Theorem 1.2.
Let L and L be links with splittings into k parts, and let C and D be partition-preserving cobordisms from L to L that are partition-homotopic. Then the maps Kh( C ) and Kh( D ) are equal up to an overall sign. The Khovanov homology of a split link satisfies a K¨unneth theorem under disjoint unions [Kho00,Proposition 33]. For simplicity, if R is a field, then for any split link L = L ∪ · · · ∪ L k , we have(1.1) Kh( L ; R ) ∼ = Kh( L ; R ) ⊗ R · · · ⊗ R Kh( L k ; R ) . Similarly, for a split cobordism C = C ∪ · · · ∪ C k between split links L = L ∪ · · · ∪ L k and L = L ∪ · · · ∪ L k , the induced maps respect the tensor product structure, in the sense that(1.2) Kh( C ) = ± Kh( C ) ⊗ · · · ⊗ Kh( C k ) . a r X i v : . [ m a t h . G T ] S e p ONKAR SINGH GUJRAL AND ADAM SIMON LEVINE
Our main technical result is that (1.2) holds not only for a split cobordism but also for an arbitrarypartition-preserving cobordism between split links. If two such cobordisms are partition-homotopic,equation (1.2) coupled with isotopy invariance of Khovanov maps then implies Theorem 1.2. (Overan arbitrary ring, there are also Tor terms in (1.1), but the same argument goes through at thechain complex level.)One consequence of Theorem 1.2 concerns cobordisms with closed components:
Corollary 1.3.
Let C : L → L be any cobordism in R × [0 , . Let S be a closed, connected,oriented surface in the complement of C ; it may be nontrivially linked with C , and may be dotted.Then: • If S is an undotted sphere, then Kh( C ∪ S ) = 0 . • If S is a dotted sphere, then Kh( C ∪ S ) = ± Kh( C ) . • If S is an undotted torus, then Kh( C ∪ S ) = ± C ) . • If g ( S ) > , or if S is dotted and g ( S ) > , then Kh( C ∪ S ) = 0 . Each of these statements was previously known in the case where S is unlinked from C (i.e.,contained in a 4-ball disjoint from C ), by results of Rasmussen [Ras05] and Tanaka [Tan06]. Wededuce the general statement by applying Theorem 1.2 to see that the map is unchanged when wepull S off of C . Thus we should interpret these statements as completing a 4-dimensional lift ofBar-Natan’s famous sphere, torus and dotted sphere relations (see Figures 2 and 3 below).1.1. Applications to ribbon concordance.
One application of Theorem 1.2, which was the orig-inal motivation behind this project, concerns strongly homotopy-ribbon concordance . A concordance from L to L is a cobordism C ⊂ R × [0 ,
1] consisting of disjoint annuli C , . . . , C k , each connectinga component of L to a component of L . Following [MZ19], a concordance C is called: • ribbon if projection onto the [0 ,
1] factor, restricted to C , is a Morse function with only index0 and 1 critical points; • strongly homotopy-ribbon if the complement of C in S × [0 ,
1] can be built from ( S (cid:114) nbd( K )) × [0 ,
1] by adding 4-dimensional 1- and 2-handles only; • homotopy-ribbon if the induced map π ( S (cid:114) L ) → π (( S × [0 , (cid:114) C ) is injective and theinduced map π ( S (cid:114) L ) → π (( S × [0 , (cid:114) C ) is surjective.Here, we have implicitly identified S with R ∪ {∞} . Gordon [Gor81] showed that the implicationsribbon ⇒ strongly homotopy-ribbon ⇒ homotopy-ribbonboth hold. (Showing injectivity on π is the most difficult piece of the argument; it relies onmajor theorems of Gerstenhaber–Rothaus [GR62] and Thurston [Thu82].) It is unknown whetherthe reverse implications hold, however. Furthermore, Gordon conjectured (and proved under someadded hypotheses) that if there are ribbon concordances from K to K and from K to K (where K and K are knots), then K and K must in fact be isotopic. Philosophically, if there is a ribbonconcordance from L to L , one expects L to be “no more complicated” than L , as measured bya variety of invariants.In 2019, Zemke [Zem19a] proved that any ribbon concordance induces an injection on knot Floerhomology. Shortly thereafter, Zemke and the second author [LZ19] proved an analogous result forKhovanov homology, using a key topological lemma proved in [Zem19a] in conjunction with someof the formal properties for Khovanov homology established by Bar-Natan [BN05]. A few monthslater, Miller and Zemke [MZ19] extended Zemke’s original result for knot Floer homology to the a priori weaker hypothesis of a strongly homotopy-ribbon concordance. Corollary 1.3 allows us toobtain the corresponding result in the Khovanov setting: Theorem 1.4.
Let S : L → L (cid:48) be a strongly homotopy-ribbon concordance. Then the map inducedon Khovanov homology Kh( S ) is injective. The definition of strongly homotopy-ribbon is purely in terms of the topology of the comple-ments of the links and the concordance, with no reference to link diagrams or movies. In contrast,
HOVANOV HOMOLOGY AND COBORDISMS BETWEEN SPLIT LINKS 3
Khovanov homology is defined entirely in terms of the combinatorics of link diagrams, and its con-nection to the topology of link complements is still largely mysterious. Theorem 1.4 is thus anunusual instance in which a non-diagrammatic property has implications for Khovanov homology.Furthermore, Theorem 1.4 implies that the various corollaries presented in [LZ19] also apply for astrongly homotopy-ribbon concordance.We also prove an additional, purely topological result concerning ribbon concordance of links:
Theorem 1.5. If L is strongly homotopy-ribbon concordant to L (cid:48) , and L (cid:48) is split, then L is split.More precisely, if there is a -sphere in S (cid:114) L (cid:48) separating two components of L (cid:48) , then there is a -sphere in S (cid:114) L separating the corresponding components of L . This follows from the injectivity results mentioned above ([LZ19] for ribbon concordance, Theorem1.4 for strongly homotopy-ribbon concordance) along with recent work by Lipshitz and Sarkar [LS19]showing that Khovanov homology detects split links. We also give a second proof using HeegaardFloer homology, making use of a similar injectivity result due to Daemi, Lidman, Vela-Vick, andWong [DLVVW20]. This result provides another example of nondecreasing simplicity under ribbonconcordance, as mentioned above.1.2.
Batson–Seed homology.
The proof of Theorem 1.2 makes use of a variant of Khovanovhomology due to Batson and Seed [BS15]. Let R be a commutative ring with unit. A weightedlink is a link L along with a choice of element of R associated to each component of L ; call thisdata w . Given a weighted link ( L, w ), Batson and Seed defined a perturbation of the differential onthe Khovanov complex, leading to a homology theory which we call Kh BS ( L, w ). The Batson–Seedcomplex is filtered, with associated graded complex equal to the original Khovanov complex (withcoefficients in R ); thus, there is a spectral sequence from Kh( L ; R ) to Kh BS ( L, w ). If L is a totallysplit link (or, more generally, a partially split link in which the weights of all components of eachpart agree), then the spectral sequence collapses, and Kh BS ( L, w ) ∼ = Kh( L ; R ). The key propertyof the Batson–Seed complex is that up to isomorphism it is unchanged (up to an overall gradingshift) upon reversing crossings between strands the difference of whose weights is invertible in R .Thus, if all such differences are invertible, the invariant Kh BS ( L, w ) depends only on the Khovanovhomology of the individual components of L .The bulk of this paper is devoted to developing a theory of tangle invariants, cobordism maps,and functoriality for Batson–Seed homology. A weighted cobordism between weighted links is a linkcobordism along with a choice of weights on each component, agreeing with the weights on theboundary. We prove: Theorem 1.6.
Let ( L , w ) and ( L , w ) be weighted links.(1) Any weighted cobordism ( C, w ) from ( L , w ) to ( L , w ) induces a filtered chain map onBatson–Seed complexes, giving an induced homomorphism Kh BS ( C, w ) : Kh BS ( L , w ) → Kh BS ( L , w ) . The induced map on associated graded objects agrees with
Kh( C ) : Kh( L ; R ) → Kh( L ; R ) .(2) If C and D are isotopic rel boundary and are equipped with the same weighting, then thefiltered chain maps associated to C and D are filtered chain homotopic up to an overallsign. Thus, Kh BS ( C, w ) and Kh BS ( D, w ) agree up to sign, as do the induced morphisms ofspectral sequences.(3) If C and D are partition-homotopic and are equipped with the same weighting, and alldifferences of weights between distinct components are invertible in R , then the chain mapsassociated to C and D are chain homotopic up to multiplication by a unit in R , althoughnot necessarily in a filtered sense. Thus, the induced maps Kh BS ( L , w ) → Kh BS ( L , w ) agree up to a unit in R . En route to proving this theorem, we also develop an analogue of the Batson–Seed complex fortangles, taking values in a category of formal curved chain complexes of diagrams, along the linesof Bar-Natan’s “tangles and cobordisms” story for Khovanov homology [BN05]. (Batson and Seed
ONKAR SINGH GUJRAL AND ADAM SIMON LEVINE
Figure 1.
The 0- and 1-resolutions of a crossing.alluded to the possibility of a such a construction in their original paper.) In the case of a cobordismbetween split links, the proof of Theorem 1.2 then follows from studying how the crossing changeisomorphisms interact with the cobordism maps, using the fact that the Khovanov and Batson–Seedinvariants agree for split links.
Remark 1.7.
We briefly discuss the “up to sign” provisos in the theorems above. Originally, thecobordism maps on Khovanov homology were only shown to be isotopy-invariant up to an overall sign[Kho06, Jac04, BN05]. Subsequent work by Caprau [Cap08] and Clark–Morrison–Walker [CMW09]modified the construction of the cobordism maps so as to eliminate the sign ambiguity. Both of theseapproaches require expanding the coefficient ring to include i (the primitive fourth root of unity)and keeping track of more topological data than just oriented cobordisms. (A recent preprint bySano [San20] provides an alternate approach that requires adjusting the signs of the maps.) Becausethe injectivity statement of Theorem 1.4 does not require pinning down the sign, we opted to stickwith the simpler framework from [BN05], at the cost of maintaining the sign indeterminacy. Organization.
In Section 2, we describe the algebraic setup that will be used throughout thepaper. In Section 3, we use this structure to define a version of the Batson–Seed complex for tangles.Sections 4 and 5 establish the invariance and functoriality of this theory, and Section 6 describesthe analogue of Batson–Seed’s crossing change isomorphisms. In Section 7, we then assemble theseingredients to prove Theorems 1.2 and 1.6, along with some generalizations to tangles. Finally, weprove the applications to concordance (Theorems 1.4 and 1.5) in Section 8.
Acknowledgments.
The authors are deeply grateful to Nathan Dowlin, Mikhail Khovanov, GageMartin, Maggie Miller, Sucharit Sarkar, Radmila Sazdanovic, and Ian Zemke for helpful conversa-tions.The first author was supported in summer 2019 by Duke University’s Program for Research forUndergraduates (PRUV). He thanks David Kraines for organising the PRUV program. He alsothanks Lenhard Ng for serving on his senior thesis committee and for his valuable comments on anearly draft of his senior thesis.The second author was partially supported by NSF Topology grant DMS-1806437.2.
Preliminaries
The Batson–Seed complex of a link.
We begin with a brief structural summary of theoriginal Batson–Seed construction. We assume that the reader is familiar with the basics of Khovanovhomology; see [BN02] for an accessible overview.Let L be a k -component, oriented link diagram with n crossings. Let R be a commutative ringwith unit, and let w = ( w , . . . , w k ) be a weight assignment of an element of R to each component of L . Let n + (resp. n − ) denote the number of positive (resp. negative) crossings. For each v ∈ { , } n ,let | v | = (cid:80) ni =1 v i , let L v be the corresponding resolution of L according to the convention shown inFigure 1, and let k v be the number of components in L v . Let V denote the R -module R [ X ] / ( X ),graded such that deg(1) = 1 and deg( X ) = −
1. For v, v (cid:48) ∈ { , } n , we say that v (cid:48) is an immediatesuccessor of v , written v (cid:108) v (cid:48) , if v (cid:48) is obtained from v by changing a single 0 to a 1.Let C = CKh( L ; R ) = CKh( L ) ⊗ Z R , the Khovanov complex of L with coefficients in R . As an R -module, we have C = (cid:76) v ∈{ , } n C v , where C v ∼ = V ⊗ k v . The complex C comes equipped with abigrading ( i, j ), where: • i denotes the homological grading, where the summand C v sits in grading | v | − n − ; HOVANOV HOMOLOGY AND COBORDISMS BETWEEN SPLIT LINKS 5 • j denotes the quantum grading, where the grading on C v comes from shifting the internalgrading on V ⊗ k v by | v | + n + − n − .Let d + : C → C denote the standard Khovanov differential, which does not depend on R or w . We do not dwell on the definition, other than to note three properties: First, d + is a sum ofterms d v,v (cid:48) + : C v → C v (cid:48) ranging over all v (cid:108) v (cid:48) , each determined by the merge and split maps in theKhovanov TQFT. Second, d + is homogeneous of degree (+1 ,
0) with respect to the ( i, j ) bigrading.Third, we have d = 0.Batson and Seed defined a second differential d − : C → C . This differential is the sum of “back-ward” terms d v (cid:48) ,v − : C v (cid:48) → C v , ranging over all v (cid:108) v (cid:48) , each of which is equal to the appropriatemerge/split map times ± ( w a − w b ), where w a and w b are the weights of the components of L at thecrossing corresponding to the index that changes from v to v (cid:48) . (We will explain the sign conventionsbelow.) This map is homogeneous of degree ( − , − d − = 0 andthat d − d + + d + d − = 0. It follows that the differential d = d + + d − satisfies d = 0. Furthermore, d is homogeneous of degree − l = j − i , which we will refer to as the Batson–Seed grading or l -grading . Observe that the l -grading on each summand C v is obtained byshifting the internal grading on V ⊗ k v by n + − n − ; we will make ample use of this fact below.While d is not homogeneous with respect to the j grading, this grading instead gives a filtrationon ( C, d ). The associated graded complex is then simply (
C, d + ), the ordinary Khovanov complexover R . We thus obtain a spectral sequence whose E page is Kh( L ; R ) and which converges to H ∗ ( C, d ). Batson and Seed prove that the filtered homotopy type of (
C, d ) is invariant (as a singly-graded R -module) under Reidemeister moves; thus, the whole spectral sequence is a link invariant.We denote the total homology of ( C, d ) by Kh BS ( L, w ).Finally, we review the crossing change map [BS15, Section 2.4]. Let L (cid:48) be obtained from L bychanging a single crossing c , and let ( C (cid:48) , d (cid:48) ) denote the corresponding Batson–Seed complex, usingthe same weights w . Batson and Seed define a chain map f : C → C (cid:48) associated to the crossingchange, which is homogeneous of degree ± l grading. If the difference of theweights of the components of L involved at c is invertible in R , then f is in fact an isomorphismof chain complexes (with inverse given up to a unit by the reverse crossing change map). If all dif-ferences of weights are invertible, it then follows that Kh BS ( L, w ) does not see the linking betweencomponents; it is isomorphic to the Khovanov homology of the totally split link obtained by sepa-rating the components of L . However, note that the crossing change map f is neither homogeneousnor filtered with respect to the j grading; it includes terms of degrees −
2, 0, and 2. Using the latterproperty, Batson and Seed were able to use the spectral sequence from Kh( L ; R ) to Kh BS ( L, w ) toobtain lower bounds on the number of crossing changes needed to split L .2.2. Homological algebra.
In order to establish functoriality for Batson–Seed homology, we willneed a localized version of the theory for tangles, taking values in a category of formal chain com-plexes akin to Bar-Natan’s work in [BN05], but with modifications motivated by the structure ofthe Batson–Seed complex as described in the previous section. The objects of study will in fact be“curved” chain complexes in which d (cid:54) = 0. Nevertheless, most standard properties from homologicalalgebra continue to hold in this context, as stated below in Remark 2.8. Throughout let R be acommutative ring with unit.We begin by reminding the reader of several definitions from [BN05] (adapted for an arbitrarybase ring R ). Definition 2.1.
A category C is R -pre-additive if the set of morphisms between any two objects isan R -module, with the property that composition of morphisms is R -bilinear. We say C is R -additive if, in addition, it has a zero object and a well-defined direct sum operation.Note that if a category C isn’t R -pre-additive to begin with, it can be made so by consideringinstead the category in which the objects are the same as those of C , but the set of morphisms fromthe object A to B is given by the free R -module generated by Mor( A, B ) of the original category C ,with composition extended bilinearly. The category thus obtained is R -pre-additive. ONKAR SINGH GUJRAL AND ADAM SIMON LEVINE
Definition 2.2.
Following [BN05, Definition 3.2], for every R -pre-additive-category C , one candefine an R -additive category Mat( C ) with objects given by formal finite direct sums of objects of C , and a morphism f from the object A = (cid:76) nj =1 A j to the object B = (cid:76) mi =1 B i given by an m × n matrix f , the ( i, j )th entry of which is a morphism A j → B i of C . Composition in Mat( C ) is givenby matrix multiplication, where instead of multiplying numbers we compose the morphisms in C . Itis easy to check that Mat( C ) is R -additive, with the zero object given by the empty direct sum. Definition 2.3.
As in [BN05, Definition 6.1], we say that an R -pre-additive category C is graded when it satisfies the following conditions:(1) For all objects A, B in C , Mor( A, B ) is a graded R -module, in a manner thatdeg( f ◦ g ) = deg( f ) + deg( g )where applicable, and the degree of the identity is always 0.(2) There is a Z -action on objects, denoted ( m, A ) (cid:55)→ A { m } ; and for any A, B ∈ Obj( C ) and m, n ∈ Z , Mor( A { m } , B { n } ) is identified with Mor( A, B ) as R -modules with a shift inthe grading so that if f ∈ Mor(
A, B ) with deg( f ) = d , then the corresponding element ofMor( A { m } , B { n } ) has degree d + n − m .As explained in [BN05, Section 6], any R -pre-additive category satisfying (1) can be upgradedto satisfy (2) by formally introducing the shifted objects. Additionally, note that if C is graded,then Mat( C ) naturally inherits the structure of a graded category, where a matrix of morphisms ishomogeneous of degree d iff all of its entries are homogeneous of degree d . Definition 2.4.
Following [BN05, Definition 3.3], for any R -additive category, we define the categoryof chain complexes Kom( C ) as follows. An object of Kom( C ) consists of a sequence (Ω r ) r ∈ Z , whereΩ r ∈ Obj( C ) and all but finitely many are 0, along with morphisms d r : Ω r → Ω r +1 satisfying d r +1 ◦ d r = 0 for all r . A morphism f : (Ω a , d a ) → (Ω b , d b ) (a chain map ) is given a family ofmorphisms f r : Ω ra → Ω rb such that d rb ◦ f r = f r +1 ◦ d ra . As in standard homological algebra, thecomposition of morphisms is defined to be ( g ◦ f ) r = g r ◦ f r . Definition 2.5.
Following [BN05, Definition 4.1], given morphisms f, g : (Ω a , d a ) → (Ω b , d b ), a homotopy between f and g is a family of morphisms h r : Ω ra → Ω r − b such that f r − g r = d r − b ◦ h r + h r +1 ◦ d ra for all r ∈ Z . We call f and g homotopic if such a homotopy exists. A morphism f : (Ω a , d a ) → (Ω b , d b ) is called a homotopy equivalence if there exists a morphism g : (Ω b , d b ) → (Ω a , d a ) such that f ◦ g and g ◦ f are each homotopic to the respective identity morphisms. The category Kom /h ( C )is defined to have the same objects as Kom( C ), with morphisms given by homotopy classes ofmorphisms in Kom( C ). Thus, a chain map f is a homotopy equivalence iff it is an isomorphism inKom /h ( C ).We will now describe a modified version of Kom that is intended to mimic the structure of theBatson–Seed complex, as described in Section 2.1. Definition 2.6.
For any graded, R -additive category C , we define a category CKom( C ) as follows.An object of CKom( C ) consists of a sequence (Ω r ) r ∈ Z , where Ω r ∈ Obj( C ) and all but finitelymany are 0, along with morphisms d r + : Ω r → Ω r +1 and d r − : Ω r → Ω r − that are each homogeneousof degree − d r +1+ ◦ d r + = 0 and d r − − ◦ d r − = 0 for all r . That is, Ω can be viewed as achain of the form · · · (cid:47) (cid:47) Ω r − d r − (cid:47) (cid:47) (cid:111) (cid:111) Ω r d r + (cid:47) (cid:47) d r − (cid:111) (cid:111) Ω r +1 (cid:47) (cid:47) d r +1 − (cid:111) (cid:111) · · · (cid:111) (cid:111) that is eventually zero on both ends. For convenience, we can combine the d r ± maps into a matrix d ± : Ω → Ω, where Ω is the formal direct sum of all Ω r , and write d = d + + d − . Note that HOVANOV HOMOLOGY AND COBORDISMS BETWEEN SPLIT LINKS 7 d = d − = 0. However, d = d + d − + d − d + need not be zero; it is a diagonal matrix consisting ofthe morphisms λ r := d r − ◦ d r − + d r +1 − ◦ d r + : Ω r → Ω r , which we call the curvature terms . (See Remark 2.9 below regarding this terminology.) We refer to r as the homological grading .Given objects Ω a , Ω b in CKom( C ), a morphism f : Ω a → Ω b (a chain map ) is given by a family ofmorphisms f r,r +2 k : Ω ra → Ω r +2 kb for all r, k ∈ Z with the property that f d = df . We interpret thelatter as a matrix equation, where f is the matrix of all the terms f r,r +2 k ; the finiteness assumptionon the objects ensures that all entries of df and f d are finite sums. (Technically, to view f as amatrix, we should include morphisms with odd homological degree shifts as well; we require theseto be 0.) The set of such chain maps is naturally an R -module. Composition of morphisms is givenby the composition of their representative matrices, which again depends on finiteness; that is,( g ◦ f ) r,r +2 k = (cid:88) l ∈ Z g r +2 l,r +2 k ◦ f r,r +2 l . The identity morphism of Ω is the direct sum of the identity morphisms of each Ω r .For a morphism f , let us write f = (cid:80) n ∈ Z f n , where f n consists of the degree- n terms of each ofthe terms f r,r +2 k (which we denote by f r,r +2 kn ). Since all the terms in d + and d − are homogeneousof the same degree, each f n individually satisfies f n d = df n . Thus, Mor(Ω a , Ω b ) has the structureof a graded R -module. We may define a grading shift operator on Obj(CKom( C )) term-wise, givingCKom( C ) the structure of a graded category as per Definition 2.3. Whenever we refer to a morphism f being homogeneous , it is with respect to this grading unless otherwise specified. Definition 2.7.
Given morphisms f, g : Ω a → Ω b in CKom( C ), a homotopy from f to g is a family ofmorphisms h r,r +2 k +1 : Ω ra → Ω r +2 k +1 b for all r, k ∈ Z , such that f − g = hd + dh (again interpreted asa matrix equation as above). If such a homotopy exists, we say f and g are homotopic. A morphism f : Ω a → Ω b is called a homotopy equivalence if there exists a morphism g : Ω b → Ω a such that f ◦ g and g ◦ f are homotopic to the respective identity maps. As above, we define CKom /h ( C ) to be thecategory whose objects are the same as CKom( C ), and whose morphisms are homotopy classes ofmorphisms in Kom( C ). Remark 2.8.
To check that CKom /h ( C ) is actually a well-defined category, we must verify thatstandard facts of homological algebra continue to hold in CKom: namely, that homotopy betweenmaps is an equivalence relation, and that composition respects this relation. These follow fromstandard homological algebra arguments, interpreting differentials, homotopies, and chain maps asmatrices. Likewise, the composition of homotopy equivalences is easily seen to be a homotopyequivalence.Additionally, note that if h n denotes the grading n part of a homotopy n from f to g (consistingof all the degree- n terms h r,r +2 k +1 n ), then h n is a homotopy from f n − to g n − . It follows that thegrading of CKom( C ) descends to CKom /h ( C ). Remark 2.9.
In our definition, CKom( C ) is a hybrid between Kom( C ) as above and the categoriesof matrix factorizations or curved chain complexes , which have appeared in various guises in theworld of knot homologies; see, e.g., [KR08, Zem19b]. However, there are a few differences. Theobjects are equipped with a Z -valued homological grading, but the morphisms (resp. homotopies)are only required to preserve (resp. shift) a Z / C ) to avoid this ambiguity. We will see thatthe Batson–Seed complex from Section 2.1, which is an honest chain complex with d = 0, can beobtained by “tensoring” together multiple formal complexes with nontrivial curvature, which mimicsthe pattern seen in [KR08] and elsewhere. ONKAR SINGH GUJRAL AND ADAM SIMON LEVINE
Note that if (Ω , d + , d − ) is a curved complex in CKom( C ), then (Ω , d + ) is an honest complex(i.e., an object in Kom( C )), since we assume that d = 0. However, a morphism f in CKom isnot necessarily a morphism in Kom, since we only require f to commute with the total differential d = d + + d − and not individually with d + and d − ; moreover, f typically does not preserve thehomological grading, which is required for morphisms in Kom. In order to extract morphisms inKom from (certain) morphisms in CKom, we now introduce a filtered version of CKom, motivatedby the j filtration of the Batson–Seed complex from the previous section.Informally, we wish to think of the “filtration grading” on a curved complex (Ω , d + , d − ) as the sumof the homological grading (denoted r above) and the internal grading coming from the category C . The differentials d + and d − are then homogeneous of degrees 0 and − d is filtered (but not homogeneous), and d + can be thought of as the differential of the associated graded complex. Of course, in our formalismof graded categories, the gradings only make sense at the level of morphisms; there isn’t actuallya notion of taking quotients to form this associated graded complex. Nevertheless, we make thefollowing definition: Definition 2.10.
Let Ω a , Ω b be objects in CKom( C ). For any chain map f : Ω a → Ω b and integer m , let f ( m ) consist of all the terms f r,r +2 kn with n + 2 k = m . We call f p -filtered if f ( m ) = 0 for all m > p .Likewise, for a homotopy h from f to g , let h ( m ) consist of all terms h r,r +2 k +1 n with n +2 k +1 = m .We call h p -filtered if h ( m ) = 0 for all m > p . If f and g are p -filtered maps and are related by a p -filtered homotopy, we say that they are p -filtered-homotopic . When p = 0, we simply say filtered rather than 0-filtered throughout.Let CKom f ( C ) denote the subcategory of CKom( C ) whose objects are the same as CKom( C ), andwhose morphisms are the filtered maps. (The composition of two filtered maps is easily seen to befiltered; more generally, the composition of a p -filtered map and a q -filtered map is ( p + q )-filtered.)Let CKom f/h ( C ) denote the category with the same objects, whose morphisms are filtered homotopyclasses of filtered maps.If a map f is homogeneous of degree p with respect to the internal grading of C , then f ( m ) canalso be seen as the sum of all terms f r,r +2 k with 2 k = m − p . In particular, if f is both homogeneousof degree p and p -filtered, then all nonzero terms in p preserve or decrease the homological grading,and f ( p ) consists of precisely those terms that preserve the homological grading. (This will be thecase with the cobordism maps considered in Section 5.) If so, we may view f ( p ) as the “associategraded” morphism of f , in the following sense: Lemma 2.11.
Let (Ω , d + , d − ) and (Ω (cid:48) , d (cid:48) + , d (cid:48)− ) be objects in CKom( C ) .(1) If f : Ω → Ω (cid:48) is a chain map that is homogeneous of degree p and p -filtered, then f ( p ) is avalid morphism in Kom( C ) from (Ω , d + ) to (Ω (cid:48) , d (cid:48) + ) .(2) If f, g : Ω → Ω (cid:48) are chain maps that are homogeneous of degree p , p -filtered, and p -filtered-homotopic, then f ( p ) and g ( p ) are homotopic (in the sense of Definition 2.5).Proof. For part (1), for each r , the portions of df and f d that go from Ω r to Ω (cid:48) r +1 are equal:(2.1) d (cid:48) r + f r,r + d (cid:48) r +2 − f r,r +2 = f r +1 ,r +1 d r + + f r − ,r +1 d r − . Since f is p -filtered, we have f r,r +2 = f r − ,r +1 = 0, so f r +1 ,r +1 d r + = d (cid:48) r + f r,r . Thus, f ( p ) is a chain map.For part (2), let h be a homotopy from f to g , satisfying f − g = d (cid:48) h + hd . Since f , g , d , and d (cid:48) are all homogeneous (of degrees p , p , −
1, and −
1, respectively), this equation still holds if wereplace h with its degree p + 1 part; thus, we may assume h is homogeneous of degree p + 1 (andstill p -filtered). Definition 2.10 implies that h r,r +2 k +1 = 0 for all k ≥
0, while h ( p ) consists of all theterms h r,r − . Looking at the terms in the relation f − g = d (cid:48) h + hd that go from Ω r to Ω (cid:48) r gives:(2.2) f r,r − g r,r = d (cid:48) + r − h r,r − + d (cid:48)− r +1 h r,r +1 + h r +1 ,r d r + + h r − ,r d r − . HOVANOV HOMOLOGY AND COBORDISMS BETWEEN SPLIT LINKS 9
The second and fourth terms on the right vanish. Thus, f ( p ) − g ( p ) = d (cid:48) + h ( p ) + h ( p ) d + , as required. (cid:3) The next lemma, which will ultimately be useful for the proof of Theorem 1.2, concerns a relatedcase in which the morphisms and homotopies are not assumed to be filtered, but the d − maps onthe individual complexes vanish. Lemma 2.12.
Let (Ω , d + , d − ) and (Ω (cid:48) , d (cid:48) + , d (cid:48)− ) be objects in CKom( C ) , and assume that d − = 0 and d (cid:48)− = 0 .(1) If f : Ω → Ω (cid:48) is a chain map that is homogeneous of degree p , then f ( p ) is a valid morphismin Kom( C ) from (Ω , d + ) to (Ω (cid:48) , d (cid:48) + ) .(2) If f, g : Ω → Ω (cid:48) are chain maps that are homogeneous of degree p , then f ( p ) and g ( p ) arehomotopic (in the sense of Definition 2.5).Proof. The proof is exactly the same as for Lemma 2.11, except that the necessary terms in (2.1)and (2.2) now vanish because d − and d (cid:48)− are both 0. (cid:3) Example 2.13.
Let Mod R denote the category of Z -graded modules over the ring R , which is an R -additive category. For a link diagram L along with a choice of weights in a ring R , the Batson–Seed complex ( C, d + , d − ) constructed in Section 2.1 can be seen as object of CKom(Mod R ). To beprecise, as before, let V = R [ X ] / ( X ), with deg(1) = 1 and deg( X ) = −
1. The complex takes theform (
C, d + , d − ), where C r = (cid:77) v ∈{ , } n : | v | = r + n − V ⊗ k v { n + − n − } . Note that the internal grading of each summand represents the l = j − i grading; the shifts agreewith the description above. The d + and d − differentials are each homogeneous of degree − d = 0; that is, the curvaturevanishes. We can reframe Batson and Seed’s proof of invariance as saying that the isomorphismclass of C in CKom f/h (Mod R ) is an invariant of the underlying weighted link; this formulation willfollow from our more abstract statement below in Section 4.2.3. The tangle cobordism category.
We now describe the tangle cobordism categories Cob ,Cob /l , Cob • , and Cob • /l , introduced by Bar–Natan [BN05, Sections 3, 6, 11.2]. The only differenceis that we will allow formal linear combinations of morphisms over an arbitrary ring R rather thanjust over Z .Let D denote a disk in R and let C be its boundary circle. Choose a base point p ∈ C , and let B be a finite set of points in C − { p } with an even number of points. A planar tangle with boundary B is a properly embedded 1-manifold in D with boundary B . A tangle cobordism from T to T (cid:48) is aproperly embedded, compact surface S ⊂ D × I with ∂S = ( T × { } ) ∪ ( B × I ) ∪ ( T (cid:48) × { } ) . A dotted cobordism is a cobordism S along with a finite set of points on the surface S . The degree of a cobordism with d dots is defined to be(2.3) deg S := χ ( S ) − | B | / − d, where χ ( S ) is the Euler characteristic of S .It is easy to verify that the degree is additive under stacking. We also note for future referencethat the degree of a saddle cobordism is −
1, and degree of a birth or death coordism is +1. (See[BN05, Exercise 6.3].) As in [BN05], we will often use the following notation to indicate cobordismsusing planar pictures:(2.4) Birth : ∅ −→
Death : −→ ∅
Saddle : −→ = 0 = 2+ = + Figure 2.
The sphere, torus, and four-tube relations in Cob /l .= 0 • = 1 • • = 0= • + • Figure 3.
The sphere, dotted sphere, two-dot, and neck-cutting relations forCob • /l ; see [BN05, p. 1493]. Definition 2.14.
Objects of both Cob ( B ) and Cob • ( B ) are pairs T { m } , where T is a planartangle with boundary B and m ∈ Z . Morphisms in Cob ( B ) (resp. Cob • ( B )) from T { m } to T (cid:48) { n } are formal R -linear combinations of cobordisms (resp. dotted cobordisms) from T to T (cid:48) , consideredup to isotopy rel boundary. Composition of morphisms is by stacking. The degree of a cobordism S ∈ Mor( T { m } , T (cid:48) { n } ) with k dots is given by deg( S ) + n − m . A linear combination of cobordismsis considered homogeneous of degree d if all the constituent cobordisms have degree d . This givesCob ( B ) and Cob • ( B ) the structure of graded categories.When B = ∅ , we may ignore the disk D and consider the circles to lie in R and the cobordismsin R × I . Definition 2.15.
Let Cob /l ( B ) (resp. Cob • /l ( B )) denote the quotient of Cob ( B ) (resp. Cob • ( B ))in which the objects are the same as those of Cob , but the morphisms are quotiented by the localrelations in Figure 2 (resp. Figure 3). Since each relation is homogeneous, each of these inherits thestructure of a graded R -additive category. Note that the local relations for Cob /l ( B ) follow fromthose for Cob • /l ( B ); as a result, the inclusion of Cob ( B ) into Cob • ( B ) descends to the /l versions.Finally, akin to Bar-Natan’s Kob( B ), we defineCKob( B ) = CKom(Mat(Cob /l ( B ))) CKob • ( B ) = CKom(Mat(Cob • /l ( B )))CKob f ( B ) = CKom f (Mat(Cob /l ( B ))) CKob f • ( B ) = CKom f (Mat(Cob • /l ( B ))) . We will sometimes omit B from the notation if it is understood from context. Note also that up tonatural isomorphism, the categories above depend only on the number of points in B , so we mayalso write CKob( k ) or CKob • ( k ) for a set B with k points. We also define CKob /h ( B ), CKob • /h ( B ),CKob f/h ( B ), and CKob f • /h ( B ) to be the homotopy categories of the above categories. HOVANOV HOMOLOGY AND COBORDISMS BETWEEN SPLIT LINKS 11
Planar algebras.
In this section we adapt Bar-Natan’s ideas of planar algebra actions (see[BN05, Section 5]) to our homological setting.
Definition 2.16.
Following [BN05, Definition 5.1], a d -input (oriented) planar arc diagram D consists of a closed disk ∆ ⊂ R ; d pairwise disjoint disks ∆ , . . . , ∆ d in the interior of ∆; basepoints p i ∈ ∂ ∆ i and p ∈ ∂ ∆, and a properly embedded, compact, (oriented) 1-dimensional submanifold A ⊂ ∆ (cid:114) int(∆ ∪ · · · ∪ ∆ d ) whose ends are disjoint from the basepoints.Given a planar arc diagram D , let B i = ∂ ∆ i ∩ A and B = ∂ ∆ ∩ A . As explained in [BN05, Section5], the planar arc diagram induces an operation D : Obj(Cob ( B )) × · · · × Obj(Cob ( B d )) → Obj(Cob ( B ))given by plugging in tangles to ∆ , . . . , ∆ d to obtain a tangle in ∆. Likewise, for tangles T i , T (cid:48) i ∈ Obj(Cob ( B i )), D gives a multi-linear operation D : Mor Cob ( B ) ( T , T (cid:48) ) × · · · × Mor
Cob ( B d ) ( T d , T (cid:48) d ) → Mor
Cob ( B ) ( D ( T , . . . , T d ) , D ( T (cid:48) , . . . , T (cid:48) d ))obtained by gluing surfaces in ∆ i × I to A × I and extending R -linearly. The same applies to thevariants Cob • and Cob • /l . These operations satisfy the identity and associativity axioms (see [BN05,p. 1465]), giving both Obj(Cob ) and Mor(Cob ) the structure of planar algebras .We now discuss the planar algebra operations on CKob and its variants. Analogous to [BN05,Theorem 2], we have: Theorem 2.17.
The collections of categories { CKob( k ) } and { CKob • ( k ) } (as well as their fil-tered versions) each have the structure of an unoriented planar algebra; moreover, the D operationspreserve homotopy of maps and homotopy equivalence of complexes.Proof. Bar-Natan’s proof extends even in our slightly modified homological setting, mutatis mutan-dis . It suffices to point out a few points.Let D be a planar arc diagram with d inputs, as above, and consider complexes (Ω i , d i, ± ) ∈ CKob( B i ). Just as in Bar-Natan’s setting, we define D (Ω , . . . , Ω d ) := (Ω , d ± ) ∈ CKob( B ) byΩ r := (cid:77) r = r + ··· + r d D (Ω r , . . . , Ω r d d )(2.5) d ± | D (Ω r ,..., Ω rdd ) := d (cid:88) i =1 ( − (cid:80) j
In (2.6), the definition of d ± depends on the order of the inputs in D — i.e.,the indexing of the input circles of D by ∆ , . . . , ∆ d . However, different orders yield canonicallyisomorphic complexes; the proof is the same as in the setting of tensor products of chain complexesover a commutative ring. Lemma 2.19.
Let D and Ω , . . . , Ω d be as in the proof of Theorem 2.17, and let Ω = D (Ω , . . . , Ω d ) .Let λ ri : Ω ri → Ω ri be the curvature of Ω i , i.e., λ ri = d r +1 i, − d ri, + + d r − i, + d ri, − . Then the curvature λ r : Ω r → Ω r is diagonal with respect to the decomposition (2.5) , and the entry λ r | D (Ω r ,..., Ω rdd ) : D (Ω r , . . . , Ω r d d ) → D (Ω r , . . . , Ω r d d ) is given by d (cid:88) i =1 D ( I r , . . . , λ r i i , . . . I r d d ) , where I ri denotes the identity morphism of Ω ri .Proof. We compute: d ∓ d ± | D (Ω r ,..., Ω rdd ) = d ∓ ◦ (cid:88) i ( − (cid:80) ji ( − (cid:80) k − j = i r j +1 D ( I r , . . . , I r i ± i , . . . , d r k k, ∓ , . . . , I r d d ) ◦ D ( I r , . . . , d r i i, ± , . . . , I r k k , . . . I r d d )The two double sums cancel out, so we deduce: d ∓ d ± | D (Ω r ,..., Ω rdd ) = d (cid:88) i =1 D ( I r , . . . , d r i ± i, ∓ ◦ d r i i, ± , . . . , I r d d ) . and hence ( d + d − + d − d + ) | D (Ω r ,..., Ω rdd ) = d (cid:88) i =1 D ( I r , . . . , d r i − i, + d r i i, − + d r i +1 i, − d r i i, + , . . . , I r d d )as required. (cid:3) The Batson–Seed complex of a tangle
In this section, we extend Batson–Seed’s variant of Khovanov homology to tangles, taking valuesin the category CKob constructed in the previous section.Throughout this section, let T be an oriented tangle, represented by a diagram in a disk ∆. Let B = ∂T . Let us assume that the diagram has n crossings, labeled c , . . . , c n , and let n + (resp. n − )denote the number of positive (resp. negative) crossings. For each v ∈ { , } n , let | v | = (cid:80) ni =1 v i , let T v be the corresponding resolution of T according to the convention shown in Figure 1, viewed asan unoriented tangle.The Batson-Seed complex will require two additional pieces of data. A weighting on T consists ofa choice w of an element of R for each component of T . (Note that components may be either arcsor circles.) We also need to specify a checkerboard shading of the tangle diagram. A checkerboardshading gives rise to a sign assignment s which assigns to each crossing c i a sign as indicated inFigure 4. This function is easily seen to satisfy Batson–Seed’s definition of a sign assignment [BS15,Section 2.2], since adjacent crossings have the same sign if and only if the diagram alternates over-under on the segment joining them. Reversing the checkerboard coloring negates s . (Batson andSeed’s definition of a sign assignment is slightly more general in the case of a disconnected diagram,but we shall restrict our attention to sign assignments arising from a checkerboard coloring.) HOVANOV HOMOLOGY AND COBORDISMS BETWEEN SPLIT LINKS 13 s ( c ) = +1 s ( c ) = − Figure 4.
The sign assignment coming from a checkerboard shading.Let T be an oriented, weighted, shaded tangle diagram, with weight w and checkerboard shad-ing/sign assignment s . We will now define a complex in CKob( B ), which we call the Batson–Seedcomplex of (
T, w ), written BS ( T, w ) or simply BS ( T ). (We suppress the shading from the notation.)The chain spaces and d + differential will not depend at all on the weighting or shading; indeed, theywill be identical (up to a slight sign modification) to Bar–Natan’s version of the Khovanov complex,thought of as an object in Kob( B ). The d − differential will depend significantly on both w and s . Definition 3.1.
The piece of BS ( T, w ) in homological grading r − n − is defined (as an object inMat(Cob ( B ))) to be:(3.1) BS ( T, w ) r − n − = (cid:77) v ∈{ , } n : | v | = r T v { n + − n − } . That is, BS ( T, w ) is just the sum of all the objects T v (for all v ∈ { , } ), each in homologicalgrading | v | − n − and with an internal shift of n + − n − . Remark 3.2.
Note that Bar–Natan [BN05, Definition 6.4] uses a shift of r + n + − n − for the r thchain space rather than n + − n − . The reason for this difference is that the degree in Bar–Natan’swork is meant to imitate the j grading from Section 2.1, whereas here the degree captures the l = j − i grading, with respect to which the Batson–Seed total differential is homogeneous. To avoidambiguity, we will refer the degree in this setting (i.e., the internal grading of the category CKob( B )as the Batson–Seed grading .We now discuss the d + and d − differentials.3.1. The d + differential. Following the template from Section 2.1, the d + differential on BS ( T, w )will be the ordinary Khovanov differential from [BN05], albeit with a slight sign modification.For each v (cid:108) v (cid:48) , let S v,v (cid:48) be the canonical saddle cobordism from T v to T v (cid:48) , viewed as a morphismbetween the corresponding summands of BS ( T, w ). As in [BN05], if v, v (cid:48) , v (cid:48) , v (cid:48)(cid:48) ∈ { , } n are vectorssuch that v (cid:108) v (cid:48) (cid:108) v (cid:48)(cid:48) and v (cid:108) v (cid:48) (cid:108) v (cid:48)(cid:48) (so that v and v (cid:48)(cid:48) differ in two indices), then we have(3.2) S v (cid:48) ,v (cid:48)(cid:48) ◦ S v,v (cid:48) = S v (cid:48) ,v (cid:48)(cid:48) ◦ S v,v (cid:48) as morphisms in Cob ( B ).We must also “sprinkle signs” — i.e., assign a sign to each edge of the cube [0 , n , such thatthe boundary of each 2-dimensional face has an odd number of minus signs. To do this, for animmediate successor pair v (cid:108) v (cid:48) , let i be the index in which v and v (cid:48) differ, and let m ( v, v (cid:48) ) be thenumber of indices j < i for which either (1) v j = 1 and c j is a positive crossing; or (2) v j = 0 and c j is a negative crossing. We then define d r − n − + : BS ( T ) r − n − → BS ( T ) r − n − +1 to be the morphismin Mat(Cob ) whose ( v, v (cid:48) ) entry is ( − m ( v,v (cid:48) ) S v,v (cid:48) when v (cid:108) v (cid:48) , and is 0 otherwise. (Note that thisis a different sign convention than in [BN05] and elsewhere; see Section 3.5.)Observe that d + is homogeneous of degree − − d + ) = 0 then follows from (3.2) together with the following simple lemma: Lemma 3.3. If v, v (cid:48) , v (cid:48) , v (cid:48)(cid:48) ∈ { , } n are vectors such that v (cid:108) v (cid:48) (cid:108) v (cid:48)(cid:48) and v (cid:108) v (cid:48) (cid:108) v (cid:48)(cid:48) , then (3.3) m ( v, v (cid:48) ) + m ( v (cid:48) , v (cid:48)(cid:48) ) (cid:54)≡ m ( v, v (cid:48) ) + m ( v (cid:48) , v (cid:48)(cid:48) ) (mod 2) . s ( c ) Positive crossing Negative crossing1 a b b − a ) a b − b − a ) − a b a − b ) a b − a − b ) Figure 5.
Batson–Seed complexes associated to the four possible orientations andshadings of a one-crossing, four-end tangle. The numbers above the resolutionsindicate the homological grading, and s ( c ) denotes the value of the sign assignmentat the crossing. Proof.
Without loss of generality, assume that v and v (cid:48) differ in index i and that v and v (cid:48) differin index j , where i < j . The crossings c k for k (cid:54) = i contribute equally to both sides of (3.3); wemust consider the contributions to m ( v (cid:48) , v (cid:48)(cid:48) ) and m ( v, v (cid:48) ) from c i . If c i is a positive crossing, thenit contributes to m ( v (cid:48) , v (cid:48)(cid:48) ) but not to m ( v, v (cid:48) ); if c i is a negative crossing, then it contributes to m ( v, v (cid:48) ) but not to m ( v, v (cid:48)(cid:48) ). In either case, the two sides differ mod 2. (cid:3) The d − differential. As in [BS15], the terms in the d − differential on BS ( T, w, s ) consist ofthe reverse cobordisms of those in d + , weighted appropriately. To be precise, for each immediatesuccessor pair v (cid:108) v (cid:48) , let S v (cid:48) ,v be the reverse of the cobordism S v,v (cid:48) from above, viewed as a saddlecobordism from T v (cid:48) to T v . The morphism d r − : BS ( T ) r → BS ( T ) r − is then defined to be the matrix of all the morphisms(3.4) ( − m ( v,v (cid:48) ) s ( c i )( w i over − w i under ) S v (cid:48) ,v , where i is the index in which v and v (cid:48) differ, and w i over and w i under are the weights of the overstrandand understrand at the crossing c i . Observe that d − is also homogeneous of degree − d − ) = 0 proceeds just as with d + , keeping track of the extra information in(3.4). (See Case 1 in the proof of [BS15, Proposition 2.2].) This completes the verification that( BS ( T, w, s ) , d + , d − ) is a valid object of CKob ( B ). Note that we do not necessarily have d = 0;we will return to this in Section 3.4. Remark 3.4.
From (3.4), we see that changing the checkerboard shading on T has the same effecton d − as multiplying all weights by −
1. Thus, when arbitrary weights are being considered, it oftensuffices to examine only one choice of shading.3.3.
Local description.
In this section we will use the planar algebra structure of CKob (fromSection 2.4) to provide a completely local description of the BS complex.Let D be a d -input oriented planar arc diagram as in Section 2.4, and let T , . . . , T d be orientedtangles that can be inserted to the inputs of D consistent with orientations. Let T = D ( T , . . . , T d )denote the resulting tangle. We say that a collection of weightings (resp. shadings) on T , . . . , T d is compatible with D if there is a weighting (resp. shading) on T that restricts to the chosen one oneach T i . Theorem 3.5.
Let D be a d -input oriented planar arc diagram, and let T , . . . , T d be oriented,weighted, shaded tangles that are compatible with D . Then BS ( D ( T , . . . , T d )) = D ( BS ( T ) , . . . , BS ( T d )) . HOVANOV HOMOLOGY AND COBORDISMS BETWEEN SPLIT LINKS 15 s ( p ) = 1 s ( p ) = − Figure 6.
Signs associated to a boundary point p ∈ B . Proof.
This proof mimics the proof of [BN05, Theorem 2], making use of the planar algebra structureof CKob discussed in Section 2.4. Just as in that proof, it suffices to assume that each tangle T i consists of two strands crossing once; the general case then follows from the associativity of theplanar algebra structure. Taking shadings and orientation into account, there are four possibilitiesfor BS ( T i ), shown in Figure 5.Equality on the level of chain spaces is immediate: each resolution of T is obtained by applying D to resolutions of T , . . . , T d in a unique way. Likewise, each of the cobordisms S v,v (cid:48) (or S v (cid:48) ,v ) inthe definition of d ± for T is obtained by extending a saddle cobordism between resolutions of some T i by the identity on the remaining tangles.It remains to verify that the signs agree. Consider the term of d + corresponding to an immediatesuccessor pair v (cid:108) v (cid:48) differing in the i th entry. By definition, the sign that appears in the differentialon BS ( D ( T , . . . , T d )) is ( − m ( v,v (cid:48) ) , while the sign in the differential D ( BS ( T ) , . . . , BS ( T d )) is,according to (2.6), ( − (cid:80) j
Theorem 3.6.
Suppose T , T are oriented, weighted, shaded tangle diagrams T , T that agree(including in their orientations, weightings, and shadings) outside a disk ∆ , and let T (cid:48) = T ∩ ∆ and T (cid:48) = T ∩ ∆ , with induced orientations, weightings, and shadings. Then:(1) For any chain map f : BS ( T (cid:48) ) → BS ( T (cid:48) ) , there is a natural extension f : BS ( T ) → BS ( T ) .(2) If f, f (cid:48) : BS ( T (cid:48) ) → BS ( T (cid:48) ) are homotopic, then so are their extensions.Proof. This follows directly from Theorem 3.5 and Theorem 2.17. To be precise, we may choosesmall disks around each of the crossings of T that are not in ∆, and view the remainder of T as aplanar arc diagram D . We then extend f by taking its planar algebra product (as in the proof ofTheorem 2.17) with the identity morphisms of the BS complexes of the single-crossing tangles. Theproof for homotopies is similar. (cid:3) Curvature terms.
With the local story in hand, we now show that the curvature terms in BS ( T ) have a particularly nice form when we make use of the dotted cobordism category.Given an oriented, weighted, shaded, oriented tangle T with boundary B , let us assign signs to theboundary points p ∈ B as follows: let s ( p ) = +1 (resp. s ( p ) = −
1) if, as we traverse the boundary ofthe disk counterclockwise, we pass from shaded to unshaded (resp. unshaded to shaded) at p , as seenin Figure 6. Let w ( p ) denote the weight of the strand containing p . Let X p : BS ( T, w ) → BS ( T, w )denote the chain map that consists of, for each v ∈ { , } n , the identity cobordism T v × [0 ,
1] witha single dot placed on the component containing p . Proposition 3.7.
Let T be an oriented, weighted, shaded tangle diagram with boundary B , andconsider the complex BS ( T ) as an object in Kob • ( B ) . Then the curvature λ T of BS ( T ) is diagonal with respect to the decomposition BS ( T ) = (cid:76) v ∈{ , } n T v , and (3.5) λ T | T v = (cid:88) p ∈ B s ( p ) w ( p ) X p . Proof.
We begin by considering the case where T is a single-crossing tangle, as shown in Figure 5.Let s ( c ) be the sign associated to the crossing by Figure 4, and let a and b denote the weights of theunderstrand and overstrand, respectively, as shown. Note that the two points in B with w ( p ) = b have s ( p ) = s ( c ), and the two points with w ( p ) = a have s ( p ) = − s ( c ).In either case, the curvature, restricted to each T v , is then equal to s ( c )( b − a ) times the compositeof two saddle cobordisms, which is obtained by tubing together the two components of T v × [0 , Cob • /l ) to the sum of thetwo ways of putting a dot on either component of T v × [0 , λ T has the desired form.For the general case, let T be a tangle diagram with n crossings. We may think of T as obtainedby inserting 1-crossing tangles T , . . . , T n into the inputs of a planar arc diagram D . By Theorem3.5, we have BS ( T ) = D ( BS ( T ) , . . . , BS ( T n )). By Lemma 2.19, the λ T is diagonal with respect tothe decomposition BS ( T ) = (cid:76) v T v , and(3.6) λ T | T v = (cid:88) p ∈ ∂T ∪···∪ ∂T n s ( p ) w ( p ) X p , where the signs s ( p ) come from the boundary orientations of the disks containing the small tangles T i . If p and p (cid:48) are points of ∂T ∪ . . . ∂T n that are connected by an arc of D , then s ( p ) = − s ( p (cid:48) ), sothe corresponding terms of (3.6) cancel. Likewise, for p, p (cid:48) ∈ ∂T , the corresponding terms in (3.5)cancel. On the other hand, if p ∈ ∂T ∪ . . . ∂T n and p (cid:48) ∈ ∂T are connected by an arc of D , then p contributes equally to the sums in (3.5) and (3.6), which completes the proof. (cid:3) As an immediate consequence of Proposition 3.7, we see that when L is a link, the total differential d on BS ( L, w ) satisfies d = 0, as shown by Batson and Seed [BS15, Proposition 2.2] for the originalversion of their chain complex. As we noted in Remark 2.9, this is akin to obtaining an honest chaincomplex from a tensor product of matrix factorizations, as in [KR08] and elsewhere.3.5. More on signs.
We now elaborate on our convention for sprinkling signs, as discussed inSection 3.1, and how it differs from the convention elsewhere in the literature. (The casual readermay safely skip this section.)In place of our m ( v, v (cid:48) ), Bar-Natan uses the quantity m (cid:48) ( v, v (cid:48) ) := (cid:80) j
Lemma 3.8.
Let m be any function on the set of all immediate successor pairs in { , } n thatsatisfies (3.3) , and let d m + and d m − be the differentials on BS ( L, w ) defined as above using the signs ( − m ( v,v ) where applicable. Then the isomorphism type of ( BS ( L, w ) , d m + , d m − ) (in CKob f ) is inde-pendent of m .Proof. Pairs v (cid:108) v (cid:48) correspond to edges of the cube [0 , n with its standard cell structure. Afunction m as in the definition may thus viewed as a cellular cochain µ m ∈ C ([0 , n ; Z / δµ m evaluates to 1 on every 2-dimensional face. If m (cid:48) isanother such function, then δ ( µ m (cid:48) − µ m ) = 0, so µ m (cid:48) = µ m + δφ for some φ ∈ C ([0 , n ; Z /
2) since H ([0 , n ; Z /
2) = 0. That is, for each v (cid:108) v (cid:48) , we have m (cid:48) ( v, v (cid:48) ) = m ( v, v (cid:48) ) + φ ( v ) − φ ( v (cid:48) ) . We then obtain an isomorphismΦ : ( BS ( L ) , d m + , d m − ) → ( BS ( L ) , d m (cid:48) + , d m (cid:48) − )consisting of ( − φ ( v ) times the identity cobordism of each resolution T v . (cid:3) Next, note that the function m depends on the orientation of T , since it is based on the signsof the crossings, while Bar-Natan’s does not. However, this dependence also disappears (up to iso-morphism) thanks to the preceding lemma. To be precise, let T and T (cid:48) denote different orientationson the same underlying unoriented (weighted, shaded) tangle diagram. Let n ± and n (cid:48)± denote thenumber of ± crossings in T and T (cid:48) , respectively. On the level of chain spaces, we have BS ( T ) = BS ( T (cid:48) )[ − n − + n (cid:48)− ] { n + − n − − n (cid:48) + + n (cid:48)− } where [ · ] represents a shift in the homological grading, and the differentials are the same up to theformula for sprinkling. By Lemma 3.8, the two complexes are isomorphic. We leave the constructionof an explicit isomorphism to the reader.Finally, while the definition of m depends on the ordering of the crossings of T , this dependenceagain disappears up to isomorphism thanks to Theorem 3.5 and Remark 2.18.3.6. Recovering the original chain complex.
To come full circle, we now discuss how to recoverBatson and Seed’s original construction from the description above.Let F : Cob • ( ∅ ) → Mod R denote the Khovanov TQFT functor, using the Frobenius algebra V = R [ X ] / ( X ). To be precise, F assigns V ⊗ k to a crossingless diagram with k circles, with agrading shift if applicable. (In Bar-Natan’s notation, v + corresponds to 1 and v − to X .) The mapsassociated to elementary birth, saddle, and death cobordisms in R × [0 ,
1] are as described in [Kho00,Section 2.2]; the map associated to a dotted cylinder is multiplication by X . The functor respectsthe relations in Figure 3, and hence descends to a functor Cob • /l ( ∅ ) → Mod R . (This is spelled outfor the undotted relations in Figure 2 in [BN05, Proposition 7.2]; the proof for the dotted relationsproceeds similarly.)If L is a weighted link diagram, applying F to BS ( L, w ) gives a curved chain complex inCKom(Mod R ), which agrees precisely with the original Batson–Seed complex, as described in Section2.1 and Example 2.13.4. Reidemeister invariance maps for the Batson–Seed complex
In this section, we prove the invariance of the BS complex up to filtered homotopy equivalence,generalizing Batson and Seed’s proof of invariance [BS15, Proposition 4.2]. More precisely, just asin [BN05, Section 4.3] (and somewhat different from [BS15]), we will explicitly construct a filteredhomotopy equivalence for each Reidemeister move. These maps will then be used in the constructionof cobordism maps in Section 5.The precise statement is as follows: Theorem 4.1.
Suppose T and T (cid:48) are oriented, weighted, shaded tangle diagrams that are relatedby a sequence of Reidemeister moves (where the weightings and shadings agree). Then there is a BS (cid:32) (cid:33) f (cid:15) (cid:15) (cid:47) (cid:47) f = − (cid:15) (cid:15) (cid:15) (cid:15) BS (cid:32) (cid:33) g (cid:79) (cid:79) g = (cid:79) (cid:79) d + = (cid:47) (cid:47) (cid:79) (cid:79) d − =0 (cid:111) (cid:111) h = (cid:106) (cid:106) Figure 7.
Reidemeister 1 invariance maps, adapted from [BN05, Figure 5]. filtered, degree- , undotted homotopy equivalence f : BS ( T ) → BS ( T (cid:48) ) whose associated graded map f (0) agrees with the homotopy equivalence on Khovanov complexes defined by Bar-Natan.Proof. Thanks to Theorem 3.6, this reduces to a local construction for each individual Reidemeistermove, which is then extended by the identity for an arbitrary tangle.For each of the three Reidemeister moves, let T and T (cid:48) denote the “before” and “after” tangles,equipped with compatible orientations, weightings, and shadings. The weights will be assumed tobe arbitrary elements of an arbitrary ring R . By Remark 3.4, since the weights are arbitrary, itsuffices to consider only one possible checkerboard shading in each case. Moreover, we will writedown the homotopy equivalences using a particular choice of orientation and ordering of crossingsfor each tangle; for an arbitrary choice of orientation and ordering of crossings, we obtain the mapby pre- and post-composing with the isomorphisms discussed in Section 3.5. We refer to (2.4) forthe symbol convention.We consider each of the Reidemeister moves separately. Reidemeister 1.
In the case of a Reidemeister 1 move, let us take T to be the 0-crossing tangle and T (cid:48) to be the 1-crossing tangle. Since the sole crossing in T (cid:48) occurs between two strands of the sametangle component, the d − differential on BS ( T (cid:48) ) vanishes, as do both the d + and d − differentials on BS ( T ). Therefore, the chain maps f : BS ( T ) → BS ( T (cid:48) ) and g : BS ( T (cid:48) ) → BS ( T ) defined by Bar-Natan [BN05], which are shown in Figure 7, are also chain maps in the sense of CKob. Moreover, g ◦ f = id, while f ◦ g is homotopic to the identity using the same homotopy h as in [BN05], whichis also shown in Figure 7. Reidemeister 2.
Similarly, in the case of a Reidemeister 2 move, let us take T to be the 0-crossingtangle and T (cid:48) to be the 2-crossing tangle. In this case, the d − differential on BS ( T (cid:48) ) may benontrivial. Let f : BS ( T ) → BS ( T (cid:48) ) and g : BS ( T (cid:48) ) → BS ( T ) be the maps defined by Bar-Natan[BN05], which are shown in Figure 8. Bar-Natan verifies that these are chain maps in the sense ofKob, which gives d + f = 0 = f d + and gd + = 0 = d + g , and an almost identical argument (using thefact that the difference of weights is the same at both crossings of T (cid:48) ) shows that d − f = 0 = f d − and gd − = 0 = d − g . Thus, f and g are also chain maps in the CKob sense. Moreover, gf = id and f g − id = d + h + hd + , where h : BS ( T (cid:48) ) → BS ( T (cid:48) ) is a homotopy shown in [BN05, Figure 6]. Itis easy to check that d − h + hd − = 0, which then shows that h is also a homotopy in the sense ofCKob. Reidemeister 3.
The case of the Reidemeister 3 move is the most interesting. Let T and T (cid:48) denote the“before” and “after” tangles, equipped with weights, shadings, orientations, and order of crossings HOVANOV HOMOLOGY AND COBORDISMS BETWEEN SPLIT LINKS 19 BS (cid:32) ab (cid:33) BS (cid:32) a b (cid:33) f (cid:15) (cid:15) g (cid:79) (cid:79) (cid:42) (cid:42) (cid:51) (cid:51) (cid:51) (cid:51) − (cid:42) (cid:42) b − a (cid:115) (cid:115) − ( a − b ) (cid:106) (cid:106) a − b (cid:106) (cid:106) b − a (cid:115) (cid:115) I (cid:17) (cid:17) I (cid:81) (cid:81) (cid:22) (cid:22) − (cid:69) (cid:69) h = − (cid:105) (cid:105) h = − (cid:91) (cid:91) Figure 8.
Reidemeister 2 invariance maps, adapted from [BN05, Figure 6]. Theweights of the two strands are given by a and b , and the small numbers inside thelower tangle indicate the order of the crossings. Each arrow in d + and d − indicatesa multiple of the appropriate saddle cobordism, with the coefficient indicated onthe arrow.as in Figure 9. Because the first two crossings (as ordered) are positive and the third is negative, ourformula for sprinkling signs agrees with Bar-Natan’s (see Section 3.5). Let f : BS ( T ) → BS ( T (cid:48) ) bethe Reidemeister 3 map constructed by Bar-Natan, which is represented by the blue arrows in Figure9. (For simplicity, we have suppressed the labels of most of the arrows in the d ± differentials on BS ( T ) and BS ( T (cid:48) ).) Bar-Natan showed that f is a chain map (in the sense of Kob) with respect tothe ordinary Khovanov differential d + ; that is, f d + = d + f . However, observe that f d − (cid:54) = d − f .For instance, if we consider only the components going from the 111 resolution of T to the 011resolution of T (cid:48) , we have f d − = 0 but d − f = b − c times a saddle cobordism.To resolve this issue, let f consist of the red arrows in Figure 9. It is easy to verify that f commutes with d − and that the failure of f to commute with d + exactly cancels the failure of f to commute with d − : f d + − d + f = d − f − f d − . For instance, going from the 111 resolution of T to the 011 resolution of T (cid:48) , we have f d + = 0 and d + f = − ( b − c ) times a saddle cobordism, which exactly cancels the contribution from d − f givenabove. The remaining components are left to the reader as an exercise; some involve using the localrelations from Figure 2. Defining f = f + f , we thus have df = f d , so f is a chain map in CKob.Note that f is homogeneous of degree 0 with respect to the Batson–Seed grading, and it is 0-filtered(in the sense of Definition 2.10) with associated graded map equal to f .To see why f is a homotopy equivalence, we first note that up to checkerboard shading andorientations, the source and target tangles of the third Reidemeister move are rotations of oneanother. Thus, up to changing the signs of some arrow, we can rotate the entire picture in Figure9, to obtain a picture describing the chain map induced by the reverse Reidemeister 3 move. It isthen a (rather tedious) exercise to see that the composition of these two maps is filtered-homotopicto the identity map of BS ( T ). (cid:3) I I I I − I ( b − c )( b − c ) Ia b ca b c b − c c − aa − b − − − −− −− − ( a − c ) c − bb − a
123 12 3 000 100 010 001 110 101 011 111
Figure 9.
The Reidemeister 3 invariance map, adapted from [BN05, Figure 9].The weights of the three tangle components are a , b , and c as shown, and thenumerals in the tangle diagrams indicate the order of the crossings. The resolutionsof T are decorated with their corresponding binary sequences. For conciseness, wehave denoted the terms in d + and d − with double-headed arrows (cf. Figures 7and 8), and we have omitted the factors s ( c i )( w over − w under ) for most of the d − terms apart from the ones in the upper left and lower right of the figure. The − signs on certain arrows denote the signs ( − m ( v,v (cid:48) ) that appear in both d + and d − differentials. (Note that the horizontal arrow in the lower-right corner includesboth this − sign and the factor of s ( c i )( w over − w under ) = a − c .) Remark 4.2.
The structure of the Reidemeister 3 map makes clear why we needed to define chainmaps in CKom to include terms with nontrivial shifts in the homological grading. Furthermore, thefact that these maps end up being homogeneous with respect to the internal grading of CKob showswhy this grading (rather than the quantum grading) is the correct choice.4.1.
Simple tangles.
Just as in [BN05], our argument will rely on showing that the BS complexesof certain simple tangles admit no non-trivial automorphisms, as we now discuss. The proofs followBar-Natan’s template nearly verbatim, but in the modified setting of our categories CKob andCKob f . Definition 4.3.
Let (Ω , d + , d − ) be an object of CKob, and consider (Ω , d + ) as an object in Kob.For C ∈ {
CKob , CKob f , Kob } , we say that Ω is C -simple if every degree-0 endomorphism of Ω in C is homotopic to a unique scalar multiple of the identity, i.e., if the map R → Mor C /h (Ω , Ω) , r (cid:55)→ [ r id Ω ]is an isomorphism.A weighted tangle ( T, w ) is called BS -simple if BS ( T, w ) is C -simple for all three choices of C .In particular, observe that any degree-0 self-homotopy equivalence of a C -simple complex mustbe (filtered) homotopic to a unit scalar multiple of the identity (cf. [BN05, Definition 8.5]).The following lemma is an easy exercise in diagram-chasing: Lemma 4.4.
Let Ω , Ω (cid:48) be objects in CKob . HOVANOV HOMOLOGY AND COBORDISMS BETWEEN SPLIT LINKS 21
T T
Figure 10.
Adding a crossing to the outside of a k -strand tangle T (left) to producea new tangle T X (right), via a planar arc diagram D k (right). (1) If Ω is C -simple, and f : Ω → Ω (cid:48) is a degree- C -homotopy equivalence (i.e., a morphism in C that is an isomorphism in C /h ), then Ω (cid:48) is C -simple.(2) If Ω and Ω (cid:48) are (filtered) simple, and f, g : Ω → Ω (cid:48) are degree- d C -homotopy equivalences,then f and g are C -homotopic up to multiplication by a unit in R . Definition 4.5. A pairing is a tangle diagram without crossings or closed components. We saythat two tangles in B are freely isotopic if they are isotopic, not necessarily rel boundary; that is,the endpoints are permitted to move. A tangle T ⊂ B is called trivial if it is freely isotopic to apairing.Because the mapping class group of an n -punctured sphere is generated by Dehn twists, twotangle diagrams represent freely isotopic tangles iff they are related by a sequence of Reidemeistermoves plus the operation of adding or removing a crossing between strands at the boundary, as inFigure 10. The main result of this section is: Lemma 4.6.
Any trivial tangle is BS -simple.Proof. The proof follows exactly along the lines of [BN05, Lemmas 8.6–8.9]. We briefly summarizethis argument. The same argument works for all three choices of C ; the case of Kob is Bar-Natan’sargument.First, we show that any pairing is BS -simple. Observe that if T is a pairing, then BS ( T ) consistsof a single tangle (namely T , without orientation) in homological grading 0, with vanishing differ-ential, so it agrees with Bar-Natan’s formal Khovanov complex Kh( T ). Moreover, any morphismin Mor CKob ( T, T ) or Mor
CKob f ( T, T ) is homogeneous with respect to the homological grading. Theproof of [BN05, Lemma 8.6] then goes through (in all three categories) to show that any such mor-phism is a multiple of the identity. For the uniqueness in Definition 4.3, first note that T can beclosed up with a planar arc diagram to obtain an unknot U . Formally, we can take a one-inputplanar arc diagram D , with no strands going to the outer boundary, such that D ( T ) = U . Then D gives a functor from CKob /h ( B ) to CKob /h ( ∅ ). The composition of this functor with the Kho-vanov TQFT takes T to Kh( U ), which a free R -module of rank 2. If r id T = 0 in Mor CKob /h ( T, T ),then r id Kh( U ) = 0, so r = 0. Thus, we deduce that Mor CKob /h ( T, T ) = R , as required. The sameargument works in CKob f .Second, we deduce from Lemma 4.4 that if T is BS -simple and T (cid:48) is obtained from T by Rei-demeister moves, then T (cid:48) is BS -simple. (See [BN05, Lemma 8.7].) The key point is that theReidemeister maps constructed above (as well as their homotopy inverses) are undotted, degree-0,filtered homotopy equivalences.Third, if T is any tangle, let T X denote the tangle obtained by adding a crossing as above,equipped with appropriate weighting and shading. Formally, we may compute BS ( T X ) from BS ( T )by “tensoring” with BS of a 1-crossing tangle, using the planar arc diagram shown in Figure 10.The argument from [BN05, Lemmas 8.7 and 8.8] applies to show that T is BS -simple if and only if T X is. Further details are left to the reader. (cid:3) Cobordism maps on the Batson–Seed complex
In this section we will establish functoriality for our Batson–Seed tangle invariant. Namely, wedescribe how weighted cobordisms between links or tangles in 4 dimensions induce filtered chainmaps between the BS complexes at either end. We then prove that these maps are invariant (up tofiltered homotopy and multiplication by a unit) under isotopy rel boundary of the cobordisms. Thisargument is essentially parallel to Bar-Natan’s argument for Khovanov homology [BN05, Section 8].5.1. Construction of the maps.
To begin, recall that any smooth cobordism C ⊂ R × [0 , L, L (cid:48) can be represented by a movie , a sequence of link diagrams L = L , . . . , L k = L (cid:48) ,where each L i is obtained from L i − by a single elementary cobordism : a Reidemeister move, birth,saddle, or death, as shown in Figure 11. Similarly, if T, T (cid:48) are oriented tangles in D with ∂T = ∂T (cid:48) ,a cobordism from T to T (cid:48) is a properly embedded, oriented surfaces C ⊂ D × [0 ,
1] with ∂C = ( − T × { } ) ∪ ( ∂T × [0 , ∪ ( T × { } ) . A tangle cobordism can likewise be described by a series of tangle diagrams in D with fixed bound-ary, each obtained from the previous one by an elementary cobordism. (See the book by Carter andSaito [CS98] for a complete exposition.) Figure 11.
Elementary cobordisms; picture taken from [BN05, Figure 10].A weighted cobordism is a link or tangle cobordism C along with a weighting w that assigns anelement of R to each component of C . A weighted movie is a movie T e −→ T e −→ . . . e k −→ T k along with weights on each each T i , such that the weights of any components in T i − and T i thatare connected by a component of e i agree. In particular, if e i is a saddle move that merges twocomponents of T i − into one component of T i or vice versa, then all three tangle components involvedmust have the same weight. Note that any movie for a weighted cobordism naturally becomes aweighted movie (where each component of each frame acquires the weight of the component of C onwhich it lies), and any weighted movie determines a weighted cobordism.For each weighted elementary cobordism e between tangles T, T (cid:48) , we define a morphism BS ( e ) : BS ( T, w ) → BS ( T (cid:48) , w (cid:48) )as follows. The tangles T and T (cid:48) are identical outside of a local “region of interest,” in which thetwo tangles differ as in Figure 11. In each case, we may define a chain map on the BS complexesof the local region (using orientation, weighting, and shading) induced from the larger tangles), andthen extend it to the larger tangles using Theorem 3.6, just like in [BN05]. More specifically:(1) For the elementary cobordism that corresponds to one of the three Reidemeister moves, wedefine the induced chain map on the BS complex to be the homotopy equivalence constructedin Section 4. HOVANOV HOMOLOGY AND COBORDISMS BETWEEN SPLIT LINKS 23 (2) For the elementary cobordism that corresponds to a birth or a death, we define the inducedchain map on the BS complex locally as the birth or death cobordism between the emptytangle and the tangle consisting of a single closed circle. In either case, the weight of theclosed circle is permitted to be an arbitrary element of R .(3) For a saddle cobordism, observe that both strands in both the source and target picturesare required to have the same weight, since they are all in the same component of thecobordism. The map BS ( T ) → BS ( T (cid:48) ) is defined to consist of a standard saddle cobordism(in D × [0 , BS ( e ) is homogeneous of degree p (with respect to the Batson–Seed grading)and p -filtered, where p equals 0 in the case of a Reidemeister move, +1 in the case of a birth ordeath, and − e in [BN05], which we will denote Kh( e ). Indeed, for all but the Reidemeister3 move, we have Kh( e ) = BS ( e ), since there are no terms that strictly decrease the homologicalgrading.Now, for a weighted cobordism C from T to T (cid:48) , we represent C by a weighted movie as above,and then define BS ( C ) to be the composition of the maps induced by the corresponding elementarycobordisms BS ( e i ). Just as in the previous paragraph, the associated graded morphism is preciselyKh( C ), as defined by Bar-Natan.5.2. Isotopy invariance.
We will now show that the BS cobordism maps constructed in the pre-vious section are invariant under isotopy rel boundary, which is precisely the content of Theorem1.6, parts 1 and 2.Just as in [Kho06, Jac04, BN05] and numerous other papers, the proof of isotopy invariancereduces to a verification for each of the movie moves of Carter and Saito [CS98], shown in Figure12. That is, two movies represent isotopic cobordisms (rel boundary) iff they differ by a sequence ofmovie moves. (To clarify the meaning of these figures, each of the movies in MM1 through MM10can be replaced by the constant movie, while each of the pairs of movies in MM11 through MM15 canbe interchanged. In each case, the movies can be read in either direction.) The technical statementof invariance is the following: Theorem 5.1.
Let L and L be weighted tangles, and let C and C (cid:48) be weighted cobordisms from L to L (cid:48) , presented by weighted movies M and M (cid:48) respectively. If M and M (cid:48) are related by a sequenceof movie moves, then BS ( M ) and BS ( M (cid:48) ) are filtered homotopic up to multiplication by ± .Proof. The argument is very similar to the argument in [BN05, Section 8]. By Theorem 3.6, itsuffices to give a local verification for each of the tangles appearing in the movie moves. Since the BS chain maps are dependent on the weighting information, this means that we need to show thatthey are unchanged (up to filtered homotopy and multiplication by ±
1) for an arbitrary weightingof each diagram that is compatible with the definition of a weighted cobordism. We now considereach of the cases: • The movies in MM1 through MM5 each consist of a Reidemeister move followed by itsinverse. As seen in Section 4, the composition is filtered homotopic to the identity, asrequired. • The movies in MM6 through MM10 are each composed entirely of Reidemeister moves, soin each case, the induced map f : BS ( T, w ) → BS ( T, w ) is a filtered, degree-0, undottedhomotopy equivalence, where (
T, w ) is the weighted tangle at the beginning and end of themovie. By Lemma 4.6, T is BS -simple, so f is filtered-homotopic to a unit scalar multipleof the identity, say r id. Therefore, the associated graded map f (0) is homotopic to r id inthe sense of Kob. By Theorem 4.1, f (0) is precisely the map on formal Khovanov complexesdefined by Bar-Natan, which is homotopic to ± id [BN05, p. 1479]. By the uniquenessprovision in our definition of BS -simplicity, we deduce that r = ±
1, as required. • For MM11, MM12, and MM13, observe the weights of all components in each tangle picturemust be the same. Hence, the d − differentials vanish, and the BS complexes and morphisms MM1 MM5MM2 MM4MM3
MM6MM10 MM7 MM9MM8
MM13 MM14 MM15MM11 MM12
Figure 12.
Movie moves; figure taken from [BN05, Figures 11–13].coincide exactly with the Kh complexes and morphisms from [BN05]. Therefore, the explicithomotopies constructed in [BN05, Page 1480] apply here as well. • In MM14 and MM15, the d − differentials may be nontrivial depending on the weights.However, because the movies are composed only of Reidemeister 2, birth, death, and saddlemoves (and no Reidemeister 3 moves), the induced chain maps coincide exactly with Bar-Natan’s maps. As seen in [BN05, p. 1481], the chain maps for the two movies in each caseare actually equal up to a sign, not just homotopic.This completes the proof. (cid:3) The Crossing Change map
In this section, we describe the analogue of the crossing change map of [BS15, Section 2.4] inpurely local terms. This will be essential for studying the effect of partition-homotopy on the mapsdescribed in the previous section.6.1.
Construction of the maps.
Let T be an oriented, weighted, shaded tangle diagram, andchoose a crossing c of T . Let T (cid:48) be the tangle that is the same as T , except that the crossing c hasbeen reversed; we may equip T (cid:48) with the same weighting and shading as T . We will define a chain HOVANOV HOMOLOGY AND COBORDISMS BETWEEN SPLIT LINKS 25 a bs ( c ) = 1 bas ( c ) = − − b − ab − a b − a b − a Figure 13.
The crossing change map. For the reverse shading, we replace b − a with a − b throughout.map CC : BS ( T, w ) → BS ( T (cid:48) , w ) , which is a chain map in the sense of Definition 2.6. Unlike the cobordism maps from Section 5, C will not be a filtered map; it may include terms that both increase and decrease the homologicalgrading. If the crossings of T are labeled c , . . . , c n and c i is the crossing being changed, we mayrefer to CC as CC ( i ) if needed. Definition 6.1.
First we specialize to the case where T consists of a single crossing c betweentwo strands, with no closed components, as in Figure 13. Let w over and w under be the weights ofthe overstrand and understrand, respectively. Note that the 0-resolution of T is identical to the1-resolution of T (cid:48) , and vice versa. We define CC to consist of the identity cobordism from T to T (cid:48) ,and s ( c )( w over − w under ) times the identity cobordism from T to T (cid:48) .In the general case, we define CC by extending the map from the previous (special) case by theidentity using Theorem 3.6.Figure 13 illustrates the local crossing change maps (in both directions) for one possible choiceof shadings. Each row shows the BS complex of a 1-crossing oriented tangle, where each arrowcorresponds to a saddle cobordism times the ring element indicated (either 1 or b − a ). The numbersabove and below the resolutions indicate the homological gradings, taking into account the shift of − n − in the definition. The upward and downward arrows give the entries in the crossing changemaps in both directions; each corresponds to the identity cobordism times the ring element indicated.To be precise, if we take T to be the upper tangle and T (cid:48) to be the lower one, we have s ( c ) = 1, w over = b , and w under = a ; while if T is the lower tangle and T (cid:48) is the upper, we have s ( c ) = − w over = a , and w under = b ; in either case, s ( c )( w over − w under ) = b − a . For the reverse shading, wesimply replace b − a with a − b in all four places where it appears, by Remark 3.4. Lemma 6.2.
For any tangle T , the map CC is a valid chain map in CKob . Moreover, if w over − w under is invertible in R , then CC is an isomorphism, with inverse given by s ( c )( w over − w under ) − times the reverse crossing change map.Proof. In the 1-crossing case, each of these properties can be verified directly from Figure 13. Namely,each crossing change map commutes with the total differential, and the composition of the crossingchange maps in both directions equals ( b − a ) times the identity.The general case then follows by applying the planar algebra structure of CKob. (cid:3) Remark 6.3.
Due to the grading shifts of n + − n − in (3.1), the map CC is of degree +2 (withrespect to the Batson–Seed grading) when a negative crossing is changed to a positive crossing,and of degree − CC consists of terms of homological degree 0 and +2 (resp. 0 and −
2) in those two cases. While CC is a filtered map in the positive-to-negative case, its inverse map is not filtered, so it is not anisomorphism in CKob f . Remark 6.4.
It is simple to describe the the crossing change map explicitly in the non-local caseas well. Note that for each v ∈ { , } n , there is a natural identification between the v -resolution of T and the v (cid:48) resolution of T (cid:48) , where v (cid:48) is obtained by changing the i th entry of v from 0 to 1 or 1to 0. The morphism CC ( i ) : BS ( T ) → BS ( T (cid:48) ) consists of the identity cobordism from T v to T (cid:48) v (cid:48) when v i = 0, and of s ( c )( w over − w under ) times the identity cobordism from T v to T (cid:48) v (cid:48) when v i = 1.This definition essentially agrees with the one given by Batson and Seed [BS15, Proof of Proposition2.3], except that the signs are different (indeed, somewhat simplified) because of our modified signconvention, as discussed in Section 3.5.6.2. Commutation of crossing change maps and cobordism maps.
Because the crossingchange maps are defined in purely local terms, the following two lemmas are immediate, using theplanar algebra structure of CKob:
Lemma 6.5.
Let T be an oriented, weighted tangle diagram. Let T (cid:48) i and T (cid:48) j be obtained by respectivelychanging the i th and j th crossings of T (where i (cid:54) = j ), and let T (cid:48)(cid:48) be obtained by changing bothcrossings. Then the following diagram commutes: BS ( T ) CC ( i ) (cid:47) (cid:47) CC ( j ) (cid:15) (cid:15) BS ( T (cid:48) i ) CC ( j ) (cid:15) (cid:15) BS ( T (cid:48) j ) CC ( i ) (cid:47) (cid:47) BS ( T (cid:48)(cid:48) ) Lemma 6.6.
Let T and T be oriented, weighted tangle diagrams that differ by a single elementarycobordism e (a Reidemeister move, birth, saddle, death). Let c i be a crossing of T that is notcontained in the region of interest of e , and let T (cid:48) and T (cid:48) be obtained from T and T by changing T i . Then the following diagram commutes: BS ( T ) BS ( e ) (cid:47) (cid:47) CC ( i ) (cid:15) (cid:15) BS ( T ) CC ( i ) (cid:15) (cid:15) BS ( T (cid:48) ) BS ( e ) (cid:47) (cid:47) BS ( T (cid:48) )Lemma 6.5 essentially says that distant crossing change maps commute with one another, whileLemma 6.6 says that crossing change maps that occur distant from an elementary cobordism mapcommute with that elementary cobordism map.In order to prove our main theorem, we will also need to understand the interaction betweenReidemeister 2 and 3 moves and crossing changes that take place within the region of interest. Let T and T (cid:48) denote the initial and final local tangles of a Reidemeister k move, where k = 2 or 3. Wemay label the strands of T by s , . . . , s k , and those of T (cid:48) by s (cid:48) , . . . , s (cid:48) k , such that s i corresponds to s (cid:48) i , and for i < j , s i crosses under s j and s (cid:48) i under s (cid:48) j .Let σ be a permutation of { , . . . , k } . Let T σ and T (cid:48) σ be obtained from T and T (cid:48) , respectively,by changing the crossing between s i and s j (for i < j ) iff σ ( i ) > σ ( j ). (In other words, we changethe order in which the strands are stacked.) It is easy to verify that T (cid:48) σ then differs from T σ by aReidemeister move of the same type, which we denote e σ . Lemma 6.7.
Let
T, T (cid:48) , T σ , T (cid:48) σ be as above, and equip all four tangles with compatible orientation,weighting, and shading. Let w i denote the weight on s i (and on the corresponding strands in theother tangles), and assume that if i < j and σ ( i ) > σ ( j ) , then w i − w j is invertible in R . Then: HOVANOV HOMOLOGY AND COBORDISMS BETWEEN SPLIT LINKS 27 (1) The maps CC : BS ( T ) → BS ( T σ ) and CC : BS ( T (cid:48) ) → BS ( T (cid:48) σ ) obtained by composing therelevant crossing change maps are each isomorphisms in CKob , and they are both homoge-neous of the same degree with respect to the Batson–Seed grading.(2) The diagram (6.1) BS ( T ) BS ( e ) (cid:47) (cid:47) CC (cid:15) (cid:15) BS ( T (cid:48) ) CC (cid:15) (cid:15) BS ( T σ ) BS ( e σ ) (cid:47) (cid:47) BS ( T (cid:48) σ ) commutes up to homotopy and multiplication by a unit.Proof. For statement 1, the CC maps are isomorphisms by Lemma 6.2 and the hypotheses, since thecrossing changes only occur between strands whose differences of weights are invertible. Moreover,the numbers of positive-to-negative and negative-to-positive crossing changes are the same for both T → T σ and T (cid:48) → T (cid:48) σ , so the degrees of the two CC maps agree by Remark 6.3. (However, notethat they are not filtered isomorphisms, or even filtered homotopy equivalences.)To prove statement 2, note that the tangles T and T (cid:48) σ are each BS -simple, by Lemma 4.6. Thecompositions BS ( e σ ) ◦ CC and CC ◦ BS ( e ) are each undotted homotopy equivalences from BS ( T )to BS ( T (cid:48) σ ), with the same quantum degree. Therefore, by part 2 of Lemma 4.4, they are homotopicup to a unit scalar, as required. (cid:3) Splitting cobordisms
Having assembled all the necessary ingredients, we now turn to the proofs of Theorem 1.2, Corol-lary 1.3, and Theorem 1.6(3). All of these will follow from a more technical statement in the worldof tangles.To begin, we describe a topological operation on tangles and movies that we call vertical separa-tion . Let T and T (cid:48) be oriented tangles with the same boundary, and let C be a tangle cobordismfrom T to T (cid:48) , represented by a movie M of the form T = T e −→ T e −→ · · · e r − −−−→ T r − e r −→ T r = T (cid:48) . Assume that we are given a decomposition C = C ∪· · ·∪ C k , where each C j is a possibly disconnectedsurface. For i = 0 , . . . , r , let T ji denote the portion of T i contained on C j , which we may think of asa tangle diagram in its own right; we refer to T i , . . . , T ki as the parts of T i . (Some of the T ji may beempty.) Let ˜ T i be the tangle diagram obtained from T i by changing crossings between T ji and T j (cid:48) i (where j < j (cid:48) ) to make the strand of T j (cid:48) i the overstrand at every crossing. Let ˜ T ji denote the partof ˜ T i obtained from T ji ; by ignoring the rest of the diagram, we see that ˜ T ji = T ji . For each j , wealso obtain a valid movie T j e j −→ T j e j −→ · · · e jr − −−−→ T jr − e jr −→ T jr (where each e ji is either an elementary cobordism or a planar isotopy) by simply erasing all but the T ji component of each from of M .Observe that for each i = 1 , . . . , r , ˜ T i is obtained from ˜ T i − by an elementary cobordism ˜ e i of thesame type as e i . This is trivial to see in the case where e i is a birth, a saddle, or a Reidemeistermove that only involves strands that all belong to the same part. In the case of a Reidemeister 2 or3 move between strands belonging to different parts, the local picture picture is just like in Lemma6.7, where we have simply permuted the heights of the strands belonging to different parts, so ˜ T i isagain obtained from ˜ T i − by a Reidemeister move. Thus, the movie(7.1) ˜ T := ˜ T e −→ ˜ T e −→ · · · ˜ e r − −−−→ ˜ T r − e r −→ ˜ T r =: ˜ T (cid:48) is a valid weighted movie, which we denote by sep( M ). See Figure 14 for an example.We may think of our tangles as lying in the 3-ball D × [0 ,
1] and our tangle cobordisms as lyingin D × [0 , × [0 , M sep( M ) split( M ) L L L L L L L L Figure 14.
A movie M for a link cobordism (left column, consisting of two Rei-demeister 2 moves followed by a saddle), and the corresponding movies sep( M )(center) and split( M ) (right).use ( x, y, z, t ) as coordinates. The tangle diagrams ˜ T i can be lifted into three dimensions in sucha way that the lift of ˜ T ji lies in D × [ j − k , jk ]. Likewise, the movie sep( M ) represents a cobordismsep( C ) = ˜ C ∪ · · · ∪ ˜ C k , where each ˜ C j lies in D × [ j − k , jk ] × [0 , C j is simply obtainedby compressing C j in the z direction to lie within the specified interval.Our main technical result says that under the right weighting hypotheses, the cobordisms C andsep( C ) induce homotopic maps on Batson-Seed complexes. Theorem 7.1.
Let ( C, w ) be a weighted tangle cobordism from T to T (cid:48) , represented by a movie M and equipped with a decomposition C = C ∪ · · · ∪ C k as above. Assume that if w j is the weight ofsome component of C j and w j (cid:48) is the weight of some component of C j (cid:48) for j (cid:54) = j (cid:48) , then w j − w j (cid:48) isinvertible in R . Give each tangle T i and ˜ T i the weighting induced from w . Then:(1) For each i = 0 , . . . , r , the morphism CC : BS ( T i ) → BS ( ˜ T i ) given as the composition of thenecessary crossing change maps is an isomorphism in CKob , and it is homogeneous withrespect to the Batson–Seed grading, with degree independent of i .(2) Each square in the diagram (7.2) BS ( T ) BS ( e ) (cid:47) (cid:47) CC (cid:15) (cid:15) BS ( T ) BS ( e ) (cid:47) (cid:47) CC (cid:15) (cid:15) . . . BS ( e r − ) (cid:47) (cid:47) BS ( T r − ) BS ( e r ) (cid:47) (cid:47) CC (cid:15) (cid:15) BS ( T r ) CC (cid:15) (cid:15) BS ( ˜ T ) BS (˜ e ) (cid:47) (cid:47) BS ( ˜ T ) BS (˜ e ) (cid:47) (cid:47) . . . BS (˜ e r − ) (cid:47) (cid:47) BS ( ˜ T r − ) BS (˜ e r ) (cid:47) (cid:47) BS ( ˜ T r ) commutes up to homotopy and multiplication by a unit, and therefore the same is true forthe composite square (7.3) BS ( T ) BS ( M ) (cid:47) (cid:47) CC (cid:15) (cid:15) BS ( T (cid:48) ) CC (cid:15) (cid:15) BS ( ˜ T ) BS (sep( M )) (cid:47) (cid:47) BS ( ˜ T (cid:48) ) . HOVANOV HOMOLOGY AND COBORDISMS BETWEEN SPLIT LINKS 29
Proof.
The hypothesis on the weights together with Lemma 6.2 guarantees that each crossing mapis an isomorphism. (Note that the order of composition of the individual crossing change maps doesnot matter, thanks to Lemma 6.5.) Moreover, it is easy to verify that the signed count of crossingsthat are changed in going from T i to ˜ T i is independent of i ; together with Remark 6.3, this showsthat the degree of the composite crossing map is independent of i as well.For statement 1, in the case where the elementary cobordism e i is is a birth, a saddle, or aReidemeister move that only involves strands that all belong to the same T ji , the square commuteson the nose by Lemma 6.6 since all crossing changes are outside the region of interest. The oneremaining case is where e i is a Reidemeister 2 or 3 move between strands belonging to different T ji ,where some crossings may have changed inside the region of the interest.Let int( T i − ), int( T i ), int( ˜ T i − ), and int( ˜ T i ) denote the intersections of each of the four tangleswith the region of interest, where int( T i − ) and int( T i ) are the before and after pictures of a Rei-demeister move e . Let ext( T i − ), ext( T i ), ext( ˜ T i − ), and ext( ˜ T i ) denote the intersections with adisk that contains all the remaining crossings, so that the complement of the two subtangles formsa planar arc diagram D . Then ext( T i − ) and ext( T i ) are identical, as are ext( ˜ T i − ) and ext( ˜ T i ),while the former is related to the latter by crossing changes. Each of the subtangles acquires itsorientation, weighting, and shading from the larger tangle. Moreover, the int tangles are relatedprecisely as in Lemma 6.7, for some permutation σ .By Theorem 3.5, we have BS ( T i ) = D ( BS (int( T i )) , BS (ext( T i ))), and so on. Moreover, thesquare from (7.2) can be broken down as D ( BS (int( T i − )) , BS (ext( T i ))) D ( BS ( e ) , id) (cid:47) (cid:47) D ( CC, id) (cid:15) (cid:15) D ( BS (int( T i )) , BS (ext( T i ))) D ( CC, id) (cid:15) (cid:15) D ( BS (int( ˜ T i − )) , BS (ext( T i ))) D ( BS (˜ e ) , id) (cid:47) (cid:47) D (id ,CC ) (cid:15) (cid:15) D ( BS (int( ˜ T i )) , BS (ext( T i ))) D (id ,CC ) (cid:15) (cid:15) D ( BS (int( ˜ T i − )) , BS (ext( ˜ T i ))) D ( BS (˜ e ) , id) (cid:47) (cid:47) D ( BS (int( ˜ T i )) , BS (ext( ˜ T i ))) . Here, the upper pair of vertical arrows are induced by the crossing changes inside the region ofinterest, while the lower pair are induced by the remaining crossing changes. The upper squarecommutes up to homotopy by Lemma 6.7, while the lower square commutes on the nose by Lemma6.6. This completes the proof. (cid:3)
We now specialize to the setting of links rather than tangles. Here, there is a further operationto consider: horizontal splitting .Let ∆ , . . . , ∆ k be disjoint disks in R . If L is a link diagram with a partition L = L ∪ · · · ∪ L k (where the parts L j may be links, or may even empty), let split( L ) be the diagram consisting ofa copy of L j inside of each ∆ j , which is uniquely determined up to planar isotopy. The diagramssep( L ) and split( L ) represent isotopic links and are related by a series of Reidemeister 2 and 3 moves;to see this, we simply slide the parts of sep( L ) apart linearly in generically chosen directions. Forany system of weights w , we have(7.4) BS (split( L ) , w ) = BS ( L , w ) ⊗ · · · ⊗ BS ( L k , w k ) , where w j denotes the restriction of w to L j , and the tensor product represents the planar algebramultiplication on Kob( ∅ ) coming from the k -input planar arc diagram with no strands.Similarly, if C = C ∪ · · · ∪ C k is a partition-preserving cobordism from L = L ∪ · · · ∪ L k to L (cid:48) = L (cid:48) ∪ · · · ∪ L (cid:48) k , represented by a movie M of the form L = L e −→ L e −→ · · · e r − −−−→ L r − e r −→ L r = L (cid:48) as above, let split( M ) be the movie consisting of a copy of the movie M j in each ∆ j . (We may alsothink of split( M ) as the split union of the movies M , . . . , M k , and denote it M (cid:116) · · · (cid:116) M k .) Again, with the identifications coming from (7.4), the map BS (split( M )) (with any choice of compatibleweights) decomposes as the product of maps: BS (split( M )) = BS ( M ) ⊗ · · · ⊗ BS ( M k ) . See Figure 14.Observe that the movies sep( M ) and split( M ) represent isotopic cobordisms in R × I . Technically,since the initial and final diagrams of M are different diagrams from the initial and final diagramsof L , this isotopy is not fixed on the boundary. The precise statement of the isotopy in terms ofmovies is thus as follows: Let N be the movie consisting of Reidemeister moves from L to split( L ) asabove, followed by split( M ) (which runs from split( L ) to split( L (cid:48) ), followed by Reidemeister movesfrom split( L (cid:48) ) to L (cid:48) . Then sep( M ) and N are related by a sequence of movie moves.From the isotopy invariance of maps, we thus deduce: Lemma 7.2.
Let L and L (cid:48) be link diagrams with decompositions L = L ∪ · · · ∪ L k and L (cid:48) = L (cid:48) ∪· · ·∪ L (cid:48) k , and let C = C ∪· · ·∪ C k be a partition-preserving cobordism from L to L (cid:48) , representedby a movie M . Let w be any choice of weights on M . Then there are homotopy equivalences makingthe following diagram commute up to filtered homotopy (up to sign): (7.5) BS (sep( L ) , w ) BS (sep( M )) (cid:47) (cid:47) (cid:39) (cid:15) (cid:15) BS (sep( L (cid:48) ) , w ) (cid:39) (cid:15) (cid:15) (cid:78) j BS ( L j , w j ) (cid:78) j BS ( M j ) (cid:47) (cid:47) (cid:78) j BS ( L (cid:48) j , w j ) . Recall that two partition-preserving cobordisms C = C ∪ · · · ∪ C k and D = D ∪ · · · ∪ D k arecalled partition-homotopic if C j is isotopic rel boundary to D j for all i . Observe that if C and D arepartition-homotopic, then sep( C ) and sep( D ) are isotopic rel boundary, as are split( C ) and split( D ). Proof of Theorem 1.6(3).
Let C and D be partition-homotopic cobordisms between partitioned links L and L as in the previous paragraph. Let w be a weighting on C , such that all differences ofweights between components in different C j are invertible in R , and let w and w denote theinduced weightings on L and L , respectively. The identification between components of C andcomponents of D induces a weighting on D , which we also denote by w . Applying Theorem 7.1 andLemma 7.2, we deduce that BS ( C ) ∼ BS (split( C )) ∼ BS (split( D )) ∼ BS ( D ) , up to multiplication by units in R , where the homotopies hold in CKom( ∅ ). Applying the KhovanovTQFT followed by the homology functor, we deduce that the mapsKh BS ( C ) , Kh BS ( D ) : Kh BS ( L , w ) → Kh BS ( L , w )are equal up to multiplication by a unit in R , as required. (cid:3) Next, we turn to Theorem 1.2, concerning partition-preserving cobordisms between split links.As in Definition 1.1, a decomposition L = L ∪ · · · ∪ L k is called a splitting if L , . . . , L k arecontained in disjointly embedded 3-balls in R . Note that this is really a property of an isotopyclass of links. Given any splitting, we may perform an ambient isotopy of R (which may be realizeddiagrammatically by Reidemeister moves) such that the 3-balls are round and centered at points inthe ( x, y ) plane. The projection of L onto this plane then yields a split diagram for L , in whichwhere the images of L , . . . , L k are contained in disjoint disks (and hence do not cross).Let L and L be links, each equipped with a k -part splitting L = L ∪ · · · ∪ L k and L = L ∪ · · · ∪ L k and represented with by split diagrams. Let C = C ∪ · · · ∪ C k be a partition-preserving cobordism. Because the diagrams are split, we have L = ˜ L and L = ˜ L . Furthermore,if we choose a weighting on C such that for each j = 1 , . . . , k , every component of C j (and hence everycomponent of L j and L j ) has the same weight w j , observe that the d − differentials on BS ( L , w )and BS ( L , w ) both vanish identically, since the only crossings are between strands with the sameweight. HOVANOV HOMOLOGY AND COBORDISMS BETWEEN SPLIT LINKS 31
Theorem 1.2 will follow from the following lemma:
Lemma 7.3.
Let L = L ∪ · · · ∪ L k and L (cid:48) = L (cid:48) ∪ · · · ∪ L (cid:48) k be split link diagrams, and let C = C ∪ · · · ∪ C k be a partition-preserving cobordism from L to L (cid:48) , represented by a movie M .Then Kh( C ) ∼ ± Kh( C ) ⊗ . . . Kh( C k ) (in Kob( ∅ ; Z ) ).Proof. Unlike in the preceding work, where the coefficient ring was arbitrary, we now wish to spe-cialize to the case where R = Z . Since this ring has very few invertible elements (namely ± k . (Note that k is simply thenumber of parts in the partition; the cobordisms C j and the links L j and L (cid:48) j need not be connected.)First, consider the base case k = 2. Assign the weight 1 to all components of C , and assign theweight 0 to all components of C . Consider the square (7.3), with L and L (cid:48) in place of T and T (cid:48) .By assumption, in the diagram L (resp. L (cid:48) ), there are no crossings between components of L and L (resp. L (cid:48) and L (cid:48) ). As a result, the vertical maps are simply the identity, and we deduce fromTheorem 7.1 that BS ( M ) ∼ ± BS (sep( M )) in CKob( ∅ ; Z ). Furthermore, the d − differentials vanishon both BS ( L ) and BS ( L (cid:48) ). By Lemma 2.12(2), we deduce that the homological grading 0 partsof BS ( C ) and BS (sep( C )) are homotopic (up to sign) in Kob( ∅ ; Z ). By construction (see Section5.1, these are Kh( C ) and Kh(sep( C )), respectively, as required. And by the isotopy invariance ofthe Khovanov maps [BN05, Theorem 4], we have Kh(sep( C )) ∼ ± Kh(split( C )). This completes the k = 2 case.For the induction, let C (cid:48) = C ∪ · · · ∪ C k − . By applying the previous case to the decomposition C = C (cid:48) ∪ C k , we deduce that Kh( C ) ∼ ± Kh( C (cid:48) (cid:116) C k ). Thus, Kh( C ) ∼ ± Kh( C (cid:48) ) ⊗ Kh( C k ). Byinduction, we have Kh( C (cid:48) ) ∼ ± Kh( C ) ⊗ Kh( C k − ), which give the necessary result. (cid:3) Proof of Theorem 1.2.
Let L and L be link diagrams, and assume that we have splittings L = L ∪ · · · ∪ L k and L = L ∪ · · · ∪ L k . We do not require L and L to be split diagrams; we simplyassume that there is a sequence S i of Reidemeister moves taking L i to a split diagram L (cid:48) i . Let C and D be partition-preserving cobordisms from L to L , represented by movies M C and M D respectively, and assume that C and D are partition-homotopic. Let M (cid:48) C (resp. M (cid:48) D ) be the moviefrom L (cid:48) to L (cid:48) given consisting of the reverse of S , followed by M C (resp. M D ), followed by S .Lemma 7.3 gives Kh( M (cid:48) C ) ∼ ± Kh( M (cid:48) D ) (as morphisms in Kob( ∅ ; Z )). By pre- and post-composingwith the homotopy equivalences induced by S and S , we deduce that Kh( M C ) ∼ ± Kh( M D ), againin the abstract sense. We then apply the Khovanov TQFT followed by the homology functor todeduce that the induced maps on Khovanov homology groups are are equal up to an overall sign, asrequired. (cid:3) Remark 7.4.
In principle, Theorem 1.2 gives an obstruction to a link L being split: if there existpartition-homotopic cobordisms C, D : L → L such that the induced maps Kh( C ) and Kh( D ) arenot equal up to sign, then L cannot be split. However, there are much more direct ways of detectingsplitness using Khovanov homology, namely the results of Lipshitz and Sarkar [LS19] discussed inthe next section.We now address Corollary 1.3. To begin, suppose S is a smoothly embedded, closed, orientablesurface in R × [0 , S as a cobordism from the empty link to itself, and noting that Kh( ∅ )is defined to be Z , the induced map is multiplication by some integer, known as the Khovanov–Jacobsson number of S , KJ ( S ). When S is connected, Rasmussen [Ras05] and Tanaka [Tan06]showed that KJ ( S ) depends only on the genus of S : it is 2 if g ( S ) = 1, and 0 otherwise. Crucially,it does not depend at all on the knotting of S . The methods of Tanaka’s paper also easily show thatfor a dotted surface S • , KJ ( S • ) equals 1 if g ( S ) = 0, and 0 otherwise. Proof of Corollary 1.3.
Let C be a cobordism from L to L (cid:48) , and let S be a closed surface (possiblydotted) in the complement of S . Let S (cid:48) denote a copy of S , translated to lie in a 4-ball disjoint from C . We may view C ∪ S and C ∪ S (cid:48) as a partition-homotopic cobordisms between the split links L ∪ ∅ and L (cid:48) ∪ ∅ . By Theorem 1.2, we haveKh( C ∪ S ) ∼ ± Kh( C ∪ S (cid:48) ) = ± KJ ( S ) Kh( C ) as required. (cid:3) Corollary 1.3 together with Tanaka’s results completely determine the Khovanov–Jacobson num-ber of a surface link , i.e., a closed surface with multiple components:
Corollary 7.5.
Let S = S ∪ · · · ∪ S k be a surface link in R × [0 , . ThenKJ ( S ) = ± (cid:89) j KJ ( S j ) = (cid:40) ± k if every component S j is a torus otherwise . Proof.
Induct using Corollary 1.3. (cid:3)
Remark 7.6.
We note that Theorem 1.2 generalises to split tangles and partition-preserving tanglecobordisms between them, where by a split tangle we mean a tangle whose connected componentscan be partitioned in a way that distinct partitions can be split from each other by the use ofReidemeister moves. The proof is identical to the proof of Theorem 1.2, and it works for the simplereason that the proof of Theorem 1.2 works at the level of abstract complexes in CKob /h , withoutpassing to homology groups.8. Applications to ribbon concordance
We now turn to the original motivation for this project: applications of Khovanov homology toribbon concordance and the more general notion of strongly homotopy-ribbon concordance.When working with maps on Khovanov homology, we must a priori take some care to distinguishbetween cobordisms in R × I and in S × I . We identify S with R ∪ {∞} . While a generic linkcobordism in S × I can be arranged to avoid {∞} × I , a generic isotopy of cobordisms might not;that is, two cobordisms in R × I may be isotopic in S × I but not in R × I . The original proofs ofisotopy invariance for the maps on Khovanov homology [Kho06, Jac04, BN05] apply only to isotopyin R × I . However, Morrison, Walker, and Wedrich [MWW19] have recently shown that isotopyinvariance also holds in S × I .The key topological fact for the proof of Theorem 1.4 is the following lemma, which is essentiallydue to Miller and Zemke [MZ19]. Lemma 8.1.
Let C ⊂ S × I be a strongly homotopy-ribbon concordance from L to L . Let C denote the mirror of C , viewed as a concordance from L to L , and let D denote the compositecobordism C ∪ L C , viewed as a concordance from L to itself. Then there exist disjointly embedded -spheres S , . . . , S k ⊂ S × I (cid:114) D , and disjointly embedded -dimensional -handles h , . . . , h k ,where e i connects D to S i and is otherwise disjoint from D ∪ S ∪ · · · ∪ S k , such that the surface D (cid:48) obtained from D ∪ S ∪ · · · ∪ S k by surgery along h , . . . , h k is isotopic (rel boundary) to L × I .Proof. By [MZ19, Lemma 3.1], the complement S × I (cid:114) nbd( D ) is built from ( S (cid:114) L ) × I byattaching the following handles:(1) n n H , . . . , H n ;(3) n more 2-handles H (cid:48) , . . . , H (cid:48) n , attached along the belt circles of H , . . . , H n respectively;(4) n i = 1 , . . . , n , let Σ i be the sphere obtained as the union of the cocore of H i and the coreof H (cid:48) i . These spheres have trivial self-intersection since they are contained in S × I .As in the proof of [MZ19, Theorem 1.2], D can be transformed into the identity cobordismby isotopy and tubing with the spheres Σ i — that is, performing surgery along an embedded 3-dimensional 1-handle joining D to Σ i . Note that D may be tubed with Σ i more than once (say k i times) during this process. In other words, we take k i parallel pushoffs of Σ i (which are disjoint sinceΣ i has trivial self-intersection) and perform the each successive tubing operation with a separatecopy. These spheres, taken over all i = 1 , . . . , n are the S , . . . , S k called for in the lemma. (cid:3) HOVANOV HOMOLOGY AND COBORDISMS BETWEEN SPLIT LINKS 33
Proof of Theorem 1.4.
Let C ⊂ S × I be a strongly homotopy-ribbon concordance from L to L .Let C , D , S , . . . , S k , h , . . . , h k , and D (cid:48) be as in Lemma 8.1. Applying the neck-cutting relationsto the tubes in D (cid:48) , we haveKh( D (cid:48) ) = (cid:88) (cid:126)e ∈{ ∅ , •} k ± Kh( D ∪ S e ∪ · · · ∪ S e k k )for some choices of signs, where S ∅ i and S • i denote S i without or with a dot, respectively. ApplyingCorollary 1.3, we have: Kh( D (cid:48) ) = ± Kh( D ∪ S • ∪ · · · ∪ S • k )= ± Kh( D ) . At the same time, we have Kh( D (cid:48) ) = ± Kh(id L ) = ± id Kh( L ) by isotopy invariance. Thus, Kh( C ) ◦ Kh( C ) = ± id Kh( L ) , which shows that Kh( C ) is injective and, indeed, left-invertible. (cid:3) We now turn to the proof of Theorem 1.5, which is a purely topological statement. To begin,let F denote the field of two elements, and let R = F [ X, Y ] / ( X , Y ). Given a link L ⊂ S andpoints p, q ∈ L , Kh( L ; F ) acquires the structure of an R -module, where the action of X (resp. Y )is given by the action of the product cobordism L × I with a dot on the component containing x (resp. Y ). Moreover, if C is a link cobordism from L to L , and p , q ∈ L and p , q ∈ L arechosen such that p and p (resp. q and q ) are on the same component of C , then the induced mapKh( C ) : Kh( L ; F ) → Kh( L ; F ) is R -linear.A recent theorem of Lipshitz and Sarkar [LS19, Corollary 1.3] shows that the R -module structureof Kh( L ; F ) detects link splitting, in the following sense: there exists an embedded 2-sphere in S (cid:114) L separating p from q if and only if Kh( L ; F ) is a free R -module, where R acts as above. Proof of Theorem 1.5.
Suppose C is a strongly homotopy-ribbon concordances from L to L , andsuppose there is a 2-sphere in S (cid:114) L separating components L i and L j . Let L i and L j be thecorresponding components of L . Choose points p ∈ L i , q ∈ L j , p ∈ L i , and q ∈ L j , and equipKh( L ) and Kh( L ) with the corresponding R -module structures; then Kh( L ; F ) is free over R . Themap Kh( C ) is R -linear and has a left inverse (which is also R -linear) by Theorem 1.4, so Kh( L ; F )injects as a direct summand of Kh( L ; F ) and is therefore a projective R -module. Since R is a localring (with unique maximal ideal ( X, Y )), every projective module is free, and thus Kh( L ; F ) is freeover R . Hence, by [LS19, Corollary 1.3], there is an embedded 2-sphere in S (cid:114) L separating p from q , as required. (cid:3) Remark 8.2.
We may give an alternate proof of Theorem 1.5 using Heegaard Floer homology.Given a link L and points p, q ∈ L , the Heegaard Floer homology of the branched double cover of L , (cid:99) HF(Σ( L )), acquires the structure of an F [ X ] / ( X )–module. Building on work of earlier work ofHedden–Ni [HN13] and Alishahi–Lipshitz [AL19], Lipshitz and Sarkar [LS19] proved that (cid:99) HF(Σ( L ))is a free F [ X ] / ( X )–module if and only if p and q are separated by a 2-sphere in S (cid:114) L . They thenused this result along with a careful analysis of the Ozsv´ath–Szab´o spectral sequence from (cid:102) Kh( L ) to (cid:99) HF(Σ( L )) [OS05] to deduce analogous detection result for Khovanov homology (both reduced andunreduced), as mentioned above.Now, if L is ribbon concordant to L , then a result of Daemi, Lidman, Vela-Vick, and Wong[DLVVW20, Theorem 1.19] shows that (cid:99) HF(Σ( L )) injects into (cid:99) HF(Σ( L )) as a summand, and it isnot hard to see that this holds at the level of F [ X ] / ( X )–modules as well. The proof of Theorem1.5 then follows along the same lines as above. References [AL19] Akram Alishahi and Robert Lipshitz,
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