Invariants of 4-manifolds from Khovanov-Rozansky link homology
IINVARIANTS OF 4-MANIFOLDS FROM KHOVANOV–ROZANSKY LINK HOMOLOGY
SCOTT MORRISON, KEVIN WALKER, AND PAUL WEDRICHAbstract. We use Khovanov–Rozansky gl N link homology to define invariants of oriented smooth 4-manifolds, as skein modules constructed from certain 4-categories with well-behaved duals.The technical heart of this construction is a proof of the sweep-around property, which makes these linkhomologies well defined in the 3-sphere.
1. IntroductionFollowing the seminal articles of Jones, Witten, and Atiyah [Jon85, Wit89, Ati88], Crane and Frenkeloutlined their vision for an algebraic construction of invariants of smooth 4-dimensional manifolds [CF94,Cra95], inspired by the initial signs of categorification they saw in Lusztig’s theory of canonical bases[Lus93]. A major milestone towards this goal was Khovanov’s celebrated categorification of the Jones poly-nomial [Kho00]—now known as
Khovanov homology —which has since been rediscovered or reconstructedin many parts of mathematics and theoretical physics, see e.g. Stroppel [Str05, Str09], Gukov–Schwarz–Vafa [GSV05], Seidel–Smith [SS06] and Abouzaid–Smith [AS19], Cautis–Kamnitzer [CK08a, CK08b], andWitten [Wit12]. Rasmussen’s construction of his slice genus bound [Ras10] demonstrates that Khovanovhomology is sensitive to 4-dimensional smooth structure and shares similarities with invariants definedusing gauge theory—two impressions that have since been supported by subsequent work, such as theunknot detection theorem of Kronheimer–Mrowka [KM11].The purpose of this article is to construct a family of triply-graded vector spaces S N ( W ; L ) , dependingon an oriented smooth 4-manifold W and a link L in its boundary, from the Khovanov–Rozansky gl N link homology theories [KR08] (which specialise to Khovanov homology at N = S × [ , ] . In the second step we use these functorial invariants to construct certain4-categories, which are the algebraic objects that encode the invariant S N ( B ; L ) for the 4-ball along withthe operations induced by gluing 4-balls. In the third step we integrate this local data over an orientedsmooth 4-manifold to produce the invariant S N ( W ; L ) , which should be thought of as the Hilbert spaceof an associated non-semisimple 4 + ϵ -dimensional TQFT.The conceptual innovation of this article is the identification of a property that ensures that a 4-category has sufficiently well-behaved duality, allowing us to integrate it over an oriented smooth4-manifold. This property, which we call the the sweep-around property , is relevant in each of the twoaxiomatizations of 4-categories with duals we describe below.The computational advance in this article is an explicit verification of this property for the 4-categoriesbuilt from Khovanov–Rozansky link homology, specifically that link cobordisms represented by moviesof the form(1.1) Date : June 9 2020. a r X i v : . [ m a t h . QA ] J un nduce identity maps on the level of link homology. For a link homology theory, this property is equivalentto functoriality under link cobordisms in S × [ , ] , which has important consequences beyond the scopeof this paper, such as the injectivity of maps induces by ribbon concordances, see Kang [Kan19]. Link homology in the 3-sphere.
In the following we give an outline of the construction. We start withthe Khovanov–Rozansky link homologies, which are categorifications of the gl N quantum link invariantsof Reshetikhin–Turaev [RT90]. These link homologies take the shape of functors (cid:26) link embeddings in R link cobordisms in R × [ , ] up to isotopy rel ∂ (cid:27) KhR N −−−−→ (cid:26) bigraded vector spaceshomogeneous linear maps (cid:27) which were constructed by Ehrig–Tubbenhauer–Wedrich in [ETW18] following earlier work on func-toriality by Bar-Natan [BN05], Clark–Morrison–Walker [CMW09], and Blanchet [Bla10], and usingtechnology developed by Robert–Wagner [RW17] and Rose–Wedrich [RW16] following Mackaay–Stošić–Vaz [MSV09], Lauda–Queffelec–Rose [LQR15], and Queffelec–Rose [QR16].It is worth emphasizing that the functors KhR N considered here are defined combinatorially andcan thus be computed algorithmically, in contrast to some earlier incarnations of Khovanov–Rozanskyhomology.The first step in our construction is to show that Khovanov–Rozansky homologies make sense as functorial invariants of links in S , rather than just in R . From the point of view of link embeddings andlink cobordisms, there is not much difference between these two cases. A generic link embedding will missthe point ∞ if we consider R = S \ {∞} and a generic link cobordism embedded in S × [ , ] will miss {∞} × [ , ] . However, the analogous statement is no longer true for isotopies of link cobordisms. Whilelink embeddings and their cobordisms can be represented by link diagrams in R and movies betweenthem, there are additional isotopies of link cobordisms in S × [ , ] , that do not exist in R × [ , ] . Inaddition to the standard Carter–Rieger–Saito movie moves [CS93, CRS97], a link homology theory that isfunctorial in S additionally has to satisfy the so-called sweep-around move (1.1), which encodes a smallisotopy of a sheet of link cobordism through ∞ × [ , ] . The central technical result that we prove in §3 isthe following. Theorem 1.1.
The Khovanov–Rozansky link homologies satisfy the sweep-around move, i.e. they associateidentity maps to link cobordisms represented by movies of the form (1.1) . This move is significantly more complex than any of the Carter–Saito movie moves because it lacksany locality after the projection to R , and thus has to be checked for any tangle T with two endpoints.We do this in §3 and thereby also demonstrate how computable cobordism maps in Khovanov–Rozanskyhomology have become. The main tool in constructing the 4-manifold invariants S N is a family of 4-categorieswith sufficiently well-behaved duals. This is in analogy with the case of quantum invariants of 3-manifolds, which—in one way or another—all depend on a suitable 3-category, such as the ribbon categoryRep ( U q ( gl N )) of finite-dimensional representations of quantum gl N .In fact, the 4-categories we construct should be thought of as categorified representation categories ofquantum gl N . They are defined to have unique 0- and 1-morphisms and • • These are related, but not identical, to categories of higher representations of categorified quantum gl N . T and T are elements of the Khovanov–Rozansky homologyKhR N ( T (cid:116) T ) of the link obtained by reflecting T and gluing it with T along their correspondingendpoints.The various ways of composing k -morphisms are purely geometric for k ≤ The skein invariant.
The construction of the 4-manifold invariant S N is most straightforward whenusing the setting of a disklike 4-category or the related notion of a lasagna algebra , a 4-dimensionalanalog of a planar algebra which we introduce in §5. Restricting to degree zero in one of the gradings,the bigraded vector space S N ( W ; L ) is constructed as a skein module (inspired by the 3-dimensionalanalogs of Conway, Przytycki [Prz91] and Turaev [Tur91]) spanned by certain decorated surfaces in W bounding L , which we call lasagna fillings , modulo skein relations imposed by the operad structure of thelasagna algebra. The bigraded vector spaces S Ni ( W ; L ) for i ≥ W is the standard 4-ball, so L is a link in the 3-sphere,then S N ( B ; L ) is isomorphic to the usual Khovanov–Rozansky homology of L , and for i > S N satisfy a gluing formula [MW12, Theorem 7.2.1] expressed in terms of atensor product over a category associated to the gluing locus. In particular S N ( B × S ; { n pts } × S ) is related to the Hochschild homology of the gl N analog of Khovanov’s arc algebra. For applications ofthe latter to link homology see Rozansky [Roz10] and Willis [Wil18] ( N = ) and Gorsky–Hogancamp–Wedrich [GHW20] ( N = ∞ ).We would like to emphasize that S N should be thought of as categorifying 3 + ϵ -dimensional skeinmodule TQFTs, see Walker [Wal], or the 3-dimensional layers of Crane–Yetter–Kauffman TQFTs [CKY97]at generic q , but not the 2 + Homotopy coherence.
Current constructions of Khovanov–Rozansky link homologies proceed viaa functorial invariant of tangles and tangle cobordisms up to isotopy, taking values in the boundedhomotopy category of an additive category; see §2. Our proof of Theorem 1.1 is stronger than necessaryin the sense that it shows that a certain equivalent reformulation of the sweep-around move holds on thechain level (i.e. not just up to homotopy) provided the tangle T is presented as a partial braid closure.It is an open question whether the Khovanov–Rozansky homologies are truncations of homotopy-coherent versions from the ∞ -category of tangles to the ∞ -category of chain complexes over the sameadditive category. If this is indeed the case, then it is plausible that our method of proof would be suitablefor an analogue of Theorem 1.1 in this setting. One should then be able to construct an A ∞ disklike4-category in the sense of [MW12, §6]: we would assign the Khovanov–Rozansky chain complex to a 4-ballwith a link in its boundary, and have k -parameter families of diffeomorphisms act via the homotopiesassociated to the corresponding higher movie moves. We would thus associate a chain complex to eachsmooth 4-manifold, rather than just its homology. enus bounds. The results in this paper hold for the ordinary Khovanov–Rozansky gl N link homologiesas well as for their GL ( N ) -equivariant and deformed versions [Lee05, Kho06, BNM06, Wu12, ETW18].In the case of links in S = ∂ B , the passage from the ordinary to deformed settings gives rise tospectral sequences that were studied in [Gor04, Ras15, Wu09, RW16]. Lobb and Wu [Lob09, Wu09],following pioneering work of Rasmussen [Ras10], showed that the associated filtrations for the genericallydeformed knot homologies in S = ∂ B contain lower bounds on the slice genus, i.e. the minimal genusof smooth surfaces in B bounding the knot. Using such invariants, Freedman–Gompf–Morrison–Walkerhave outlined a strategy for testing counterexamples to the smooth 4-dimensional Poincaré conjecture[FGMW10]. One motivation for studying 4-manifold invariants from Khovanov–Rozansky homologies isthat analogous spectral sequences might give rise to lower bounds on the genera of smooth surfaces in4-manifolds W bounding knots in M = ∂ W . Relations to other work.
There have been several proposed approaches to constructing homologytheories for links in 3-manifolds, or 4-manifold invariants, which either intended to categorify sl or gl N quantum invariants or to directly generalize Khovanov–Rozansky homology. These include(1) categorifying Witten–Reshetikhin–Turaev invariants at roots of unity, see e.g. Khovanov [Kho16],Qi [Qi14], Elias–Qi [EQ16] and Qi–Sussan [QS17],(2) using 2-representations of categorified quantum groups in the sense of Rouquier [Rou08] andKhovanov–Lauda [KL10] to construct a 4-category that can be integrated over 4-manifolds, seee.g. Webster [Web17] for categorified tensor products,(3) categorifying skein algebras and 3-manifold skein modules, see Asaeda–Przytycki–Sikora [APS04]Thurston [Thu14] and Queffelec–Wedrich [QW18a, QW18b], starting from the thickened an-nulus, see Grigsby–Licata–Wehrli [GLW18], Beliakova–Putyra–Wehrli [BPW19] and Queffelec–Rose [QR18], or connect sums of S × S , see Rozansky [Roz10] and Willis [Wil18].(4) giving a mathematically rigorous construction of the BPS spectra (“relative Gromov–Witten invari-ants”) proposed by Gukov–Putrov–Vafa [GPV17] and Gukov–Pei–Putrov–Vafa [GPPV17] based onGukov–Schwarz–Vafa [GSV05], see e.g. Gukov–Manolescu [GM19] and Ekholm–Shende [ES19],(5) extending Witten’s gauge-theoretic interpretation of Khovanov homology [Wit12] from R toother 3-manifolds, see also Taubes [Tau13, Tau18].Comparing these approaches with the invariants defined in the present article may be an interesting topicfor further research. We expect a close relationship with approach (2) already at the level of 4-categories,and with approach (3) since it uses the same underlying combinatorics. The latter is especially appealingsince (3) is, on the one hand, computationally well-developed for thickened surfaces, but, on the otherhand, poses many open questions about the categorification of skein algebras and related quantum clusteralgebras, onto which our invariants might shed new light. Acknowledgements.
The authors would like to thank Ian Agol, Chris Douglas, Mike Freedman, MarcoMackaay, Anton Mellit, Stephen Morgan, and Hoel Queffelec for helpful conversations. Scott Morrisonwas partially supported by Australian Research Council grants ‘Low dimensional categories’ DP160103479and ‘Quantum symmetries’ FT170100019. Paul Wedrich was supported by Australian Research Councilgrants ‘Braid groups and higher representation theory’ DP140103821 and ‘Low dimensional categories’DP160103479. 2. TechnologyThe purpose of this section is to survey the technology used in functorial Khovanov–Rozansky linkhomologies and to set up notation. .1. Webs.
The category Rep ( U q ( gl N )) of finite-dimensional U q ( gl N ) -modules is a ribbon category andthus provides Reshetikhin–Turaev invariants of framed oriented tangles with components labeled byobjects of Rep ( U q ( gl N )) . A framed oriented link L labeled by the U q ( gl N ) -module V = C N ( q ) yields anendomorphism of C ( q ) , the tensor unit in Rep ( U q ( gl N )) , which is just multiplication by the gl N linkpolynomial of L .While we will focus on invariants of links labeled by V , it is convenient to also consider the fundamentalmodules (cid:211) k ( V ) and their duals. Together, these generate the full monoidal subcategory Fund ( U q ( gl N )) ,which admits a graphical presentation and which recovers Rep ( U q ( gl N )) upon idempotent completion. Definition 2.1.
The C ( q ) -linear pivotal category Web N has objects given by finite sets of points inan interval [ , ] , each labeled by an element of { n , n ∗ | n ∈ Z > } . The morphisms are Z [ q ± ] -linearcombinations of webs: oriented trivalent graphs, properly embedded in [ , ] , with edges labeled by anon-negative integer flow, considered up to isotopy relative to the boundary and local relations (2.1).The source and target of a web are determined by its intersections with [ , ] × { } and [ , ] × { } , withdownward orientied boundary points of label n being recorded as n ∗ . Composition is given by the bilinearextension of stacking webs and the tensor product is given on objects by concatenating labeled intervalsand on morphisms by the bilinear extension of placing webs side by side.The morphisms in Web N are generated under composition, tensor product, and duality by identitymorphisms and trivalent merge and split vertices: a , a + ba b , a + ba b The merge and split vertices encode the natural U q ( gl N ) -intertwiners (cid:211) a ( V ) ⊗ (cid:211) b ( V ) → (cid:211) a + b ( V ) and (cid:211) a + b ( V ) → (cid:211) a ( V ) ⊗ (cid:211) b ( V ) respectively. The local relations in Web N include aa − b b = (cid:2) ab (cid:3) a , aa + b b = (cid:2) N − ab (cid:3) a , a b c = a b c , k rs l = (cid:205) t (cid:2) k − l + r − st (cid:3) ks − tr − tl (2.1)together with the reflections of these relations in a vertical line. Edges labeled zero are to be erased andedges labeled by negative integers force the morphism to be the zero morphism.The relations ensure that Web N (cid:27) Fund ( U q ( gl N )) are C ( q ) -linear pivotal categories [CKM14, TVW17].In the following, we also consider an integral version of Web N , which is defined over Z [ q ± ] , subject tothe same relations (2.1).2.2. Foams.
Foams provide a framework for a combinatorial description of Khovanov–Rozansky linkhomologies, in a similar way as webs are useful for the type A Reshetikhin-Turaev invariants. We will use gl N -foams constructed via the combinatorial evaluation formula for closed foams due to Robert–Wagner[RW17]. More precisely, we will organise these gl N -foams into a monoidal bicategory Foam N whichcategorifies the integral form of Web N . Definition 2.2.
The graded, additive monoidal bicategory
Foam N has objects given finite sets of points in [ , ] , each labeled by an element of { n , n ∗ | n ∈ Z > } . The 1-morphisms are (formal direct sums of formalgrading shifts of) webs, properly embedded in [ , ] and connecting boundary points of appropriatelabels. Note that webs are not considered up to any relations in Foam N . The 2-morphisms are (matricesof degree zero) Z -linear combinations of gl N -foams in [ , ] , considered up to isotopy relative to theboundary and certain local relations, as defined in [ETW18, Section 2]. he three compositions are given by (the bilinear extension of) stacking these topological objectsalong the three interval directions.Foams are the natural notion of cobordisms between webs and the relations between 2-morphismsin Foam N are chosen such that the defining web equalities (2.1) in Web N can be lifted to explicit webisomorphisms in Foam N . We refer to [ETW18] for a rigorous definition of gl N -foams, as well as a completedescription of the relations between them, and a survey of various flavors of Foam N . Here we onlycomment on aspects relevant to the rest of this paper. Figure 1.Foams are represented by 2-dimensional cell complexes, such that everypoint has a neighborhood either modelled on R , three half-planes meetingin a line, or the cone on the 1-skeleton of a tetrahedron. Such cone pointsare called singular vertices of the foam. The points on the line in the secondcase form a seam of the foam, and the connected components of the set ofmanifold points are called the facets of the foam. An example of a foam withsix singular vertices is shown in Figure 1. The facets are oriented and labeledby positive integers. If three facets meet along a seam, then two of theirlabels, say a and b , sum to the third, a + b . The orientation of the seam agreeswith the orientation induced by the a and b facets, and disagrees with the a + b facet.Each facet of a foam in Foam N admits an action of the algebra of symmetric functions Λ . This is to saythat facets may be decorated by points labeled by symmetric functions, which are allowed to move freelyon facets. A point labeled by a product f д ∈ Λ may be split into two points labeled f and д respectively,and a foam with a point labeled f + д ∈ Λ may be split into a sum of foams with points labeled f and д respectively. The Λ -actions on adjacent facets are compatible in the sense that f ∈ Λ on an a + b facetmay be moved across a seam, where it distributes into ∆ ( f ) ∈ Λ ⊗ Λ acting on the adjacent a and b facets.The degree of a foam is computed as twice the degree of the symmetric function decoration, minus aweighted Euler characteristic, depending on facet labels. Foam N is designed to have finite-dimensional spaces of 2-morphisms, and in particular, the Λ -actionon each a -facet factors through a finite-dimensional quotient, namely H ∗ ( Gr ( C a ⊂ C N )) , the cohomologyring of the Grassmannian of a -dimensional subspaces of C N , which is obtained as quotient of Λ bythe ideal (cid:104) h N − a + i | i > (cid:105) generated by sufficiently large complete symmetric functions. In the case of a1-labeled facet, the symmetric function e = h is called the dot . Example . The algebra of decorations on a 1-facet in
Foam N can be realised as the space of 2-morphisms A = Foam N (∅ , (cid:13) ) between the empty web and a 1-labeled circle. It is spanned by foams consisting ofdisks, decorated by a number 0 ≤ n ≤ N − X n . The multiplication of suchfoams is realised by gluing two such dotted disks onto the legs of a pair of pants, giving m ( X n , X n ) = X n + n , subject to the relation that X N − + i = i >
0. In fact, A is a commutative Frobenius algebra,with counit given by capping disks off:(2.2) = (cid:213) a + b = N − • a • b , • n = δ n , N − . Thus we have A (cid:27) Z [ X ]/(cid:104) X N (cid:105) (cid:27) H ∗ ( C P N − ) as commutative Frobenius algebras, and the 1-labeled partof Foam N is nothing but the quotient of the linearised 2-dimensional oriented cobordism category by therelations in the kernel of the ( + ) -dimensional TQFT corresponding to H ∗ ( C P N − ) . More generally, wehave A k def = Foam N (∅ , (cid:13) k ) (cid:27) H ∗ ( Gr ( C a ⊂ C N )) (cid:27) (cid:211) k A and Foam N can be considered as the universal ource for a TQFT-like functor defined on foams, which evaluates to A on 1-circles and is compatiblewith induction and restriction between tensor products of exterior powers of A . Remark . There is also an equivariant version of
Foam N , with facet algebras given by the GL ( N ) -equivariant cohomology rings H ∗ GL ( N ) ( Gr ( C a ⊂ C N )) , defined over the base ring H ∗ GL ( N ) ( point ) . Thisversion is important due to its role in the proof of functoriality of Khovanov–Rozansky homology[ETW18] and as the source of Lee-type deformation spectral sequences [Lee05, RW16] and Rasmussen-type invariants [Ras10]. Everything in this paper works, mutatis mutandis, in the equivariant framework.2.3. Khovanov–Rozansky homology.
The construction of Khovanov–Rozansky link homologies nowproceeds in two steps. The first step is a functor that sends link diagrams to chain complexes in
Foam N andlink cobordisms to chain maps, which depend only on the isotopy type of the cobordism up to homotopy.The second step evaluates such a chain complex to a bigraded vector space through a representablefunctor and taking homology. Definition 2.5.
The category R Link ◦ has objects given by embedded, framed oriented links in L ⊂ R ,such that the projection along the z -axis maps L to a blackboard-framed link diagram in R × { } ⊂ R ,together with an ordering of the finitely many crossings in the diagram. The morphisms are orientedlink cobordisms in R × [ , ] up to isotopy rel boundary, together with formal crossing reorderingisomorphisms.In one direction, by forgetting the condition on the projection and ignoring the crossing order, thiscategory is equivalent to the usual category of all embedded, framed oriented links and link cobordisms.In the other direction, the category R Link ◦ is equivalent to the category whose objects are link diagramsand whose morphisms are sequences of Reidemeister moves, Morse moves, planar isotopies, and formalreorderings, considered up to Carter–Rieger–Saito movie moves [CS93, CRS97].We will now describe the construction of a functor (cid:110) − (cid:111) : R Link ◦ → K b ( Foam N ) , with target givenby the bounded homotopy category of Foam N . In particular, the functor sends link diagrams to certainbounded chain complexes of webs and foams. On single, 1-labeled crossings, it is defined as:(2.3) (cid:22) (cid:23) = q → (cid:58)(cid:58)(cid:58)(cid:58) , (cid:22) (cid:23) = (cid:58)(cid:58)(cid:58)(cid:58) → q − The (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) underlined term is placed in homological degree zero. We call the non-identity webs that appear here thick edges . The differentials in both complexes are given by the combinatorially simplest foam betweenthe two shown webs. We call them unzip and zip foams respectively.A link diagram with several crossings (in a specified order) is sent to the chain complex constructedfrom the formal tensor product of the crossing complexes (2.3) (in that order) by gluing its resolutionsinto the link diagram in place of the original crossings.The chain complexes associated to link diagrams which differ only by Reidemeister moves arehomotopy equivalent, see Sections 3.3–3.5. Similarly, one can define chain maps for Morse moves.However, a highly non-trivial fact is that there exists a coherent choice for such chain maps.
Theorem 2.6 ([ETW18]) . The construction (cid:110) − (cid:111) : R Link ◦ → K b ( Foam N ) is functorial. In fact, Theorem 2.6 holds in much greater generality, including colored links and the equivariantframework mentioned in Remark 2.4. More importantly for us, the theorem holds locally, i.e. for tanglediagrams and tangle cobordisms. efinition 2.7. The Khovanov–Rozansky gl N link homology KhR N : R Link ◦ → gr Z × Z Vect is definedas the composition of (cid:110) − (cid:111) and H ∗ ( (cid:201) k ∈ Z Foam N ( q − k ∅ , −)) . It is functorial by Theorem 2.6. Remark . In Definition 2.7 we have use the identification K b ( End
Foam N (∅)) (cid:27) K b ( gr Z Vect ) (cid:27) gr Z × Z Vect and consider the latter as a symmetric monoidal dg category with vanishing differential. Throughout thepaper, every reference to bigraded vector spaces is interpreted in this sense.3. The sweep-around moveThe purpose of this section is to prove Theorem 1.1.3.1.
Reduction to almost braid closures.
Given a braid word β for a braid [ β ] ∈ Br n + , we can get a1-1-tangle diagram by taking the braid closure of the n rightmost strands. We say that such 1-1-tanglediagrams are in almost braid closure form . From a 1-1-tangle diagram T , one can obtain link diagrams L and L (cid:48) by either taking the left- or right-handed closure of the single open strand. These diagrams areillustrated at the top left and top right of (3.1) respectively.We note the following straightforward extension of the Alexander theorem. Lemma 3.1.
Every 1-1-tangle can be isotoped into almost braid closure form.
Proposition 3.2.
If the sweep-around map is homotopic to the identity for 1-1-tangles in almost braidclosure form, then the same is true for all 1-1-tangle diagrams.Proof.
Consider an isotopy that brings the tangle diagram T into almost braid closure form T (cid:48) and denoteits image under the Khovanov invariant as ϕ . Furthermore, let the maps associated to the sweep-aroundfor T and T (cid:48) be denoted by sw T and sw T (cid:48) respectively. Now, note that sw T (cid:39) ϕ − ◦ sw T (cid:48) ◦ ϕ becausethe underlying link cobordisms are isotopic in R × [ , ] . By assumption sw T (cid:48) (cid:39) id T (cid:48) and thus alsosw T (cid:39) id T . (cid:3) The game plan.
Fix an almost closure T of a braid word β for [ β ] ∈ Br n + . We call the right-handclosure L and the left-hand closure L (cid:48) . We will consider the following movies of intermediate diagramsand their associated chain maps between Khovanov–Rozansky complexes. In the first row, the ± signsindicate the two versions of this movie, in which the horizontal strand passes in front of ( + ) or behind ( − ) T .(3.1) β β β · · · β β β (cid:18) L + (cid:19) (cid:18) L + (cid:19) · · · (cid:18) L x − + (cid:19) (cid:18) L x + (cid:19) (cid:110) L (cid:111) (cid:110) L (cid:48) (cid:111) (cid:18) L − (cid:19) (cid:18) L − (cid:19) · · · (cid:18) L x − − (cid:19) (cid:18) L x − (cid:19) . R ± R ± R ± R ± R − ± R − ± R + R + R + R − + R − + R + R − R − R − R − R − R − − We denote the composition along the top sw + and the composition along the bottom sw − . Our goalis to show that, after making careful use of the freedom, described later, to choose up-to-homotopyrepresentatives of the chain maps for Reidemeister III moves, we have the following: Theorem 3.3.
For every almost braid closure diagram T , the front sweep sw + and the back sweep sw − chainmaps constructed above are identical (not just merely homotopic). ogether with Proposition 3.2, this will imply Theorem 1.1. Corollary 3.4.
For every almost braid closure diagram T , we have sw T = T .Proof. We have sw T = ( sw − ) − ◦ sw + = ( sw − ) − ◦ sw − = T . (cid:3) The proof of Theorem 3.3 will occupy the rest of this section.We distinguish two types of crossings in the intermediate diagrams L i ± . The crossings of the moving,horizontal, strand with everything else will be called external . The remaining crossings were alreadypresent in T and will be called internal . Definition 3.5.
The homological grading on (cid:18) L i ± (cid:19) splits into the sum of the internal and external homo-logical grading , contributed by resolutions of internal and external crossings respectively. The internaland external homological degrees of a web W appearing (cid:18) L i ± (cid:19) will be denoted by gr int ( W ) and gr ext ( W ) respectively.The braid word β determines an ordering of the crossings in T , L , and L (cid:48) , namely from top to bottom .This ordering also induces an ordering of the internal crossings in all other diagrams in (3.1). The diagrams L ± and L x ± have one additional external crossing. The diagrams L i ± for 1 ≤ i ≤ x − n + W in each of these complexesaccording to the resolutions that appear at the crossings. For the following, let M denote the number ofcrossings in T , L , and L (cid:48) . Definition 3.6.
The type of a web W in any of the complexes in (3.1) is the element τ ( W ) ∈ { p , t } M thatrecords in the j -th coordinate whether the j -th internal crossing in the respective link diagram is resolvedin a parallel way (p), or using the thick edge (t).The offset of a web W in any of the complexes (cid:18) L i ± (cid:19) is the element o ( W ) ∈ { p , t } that records theresolution of the leftmost external crossing.The state of a web W in any of the complexes (cid:18) L i ± (cid:19) for 1 ≤ i ≤ x − s ( W ) ∈ { p , t } n ,which records the resolutions of the 2 n rightmost external crossings (that is, all except the leftmostexternal crossing). Such a web W is said to be palindromic if s ( W ) is a palindrome. Lemma 3.7.
The webs W in the complexes (cid:110) L (cid:111) and (cid:110) L (cid:48) (cid:111) are indexed by their types τ ( W ) . The webs W in (cid:18) L ± (cid:19) , and (cid:18) L x ± (cid:19) are indexed by the pairs ( τ ( W ) , o ( W )) . The webs W in the complexes (cid:18) L i ± (cid:19) for ≤ i ≤ x − are indexed by the triples ( τ ( W ) , o ( W ) , s ( W )) . Definition 3.8. If i ∈ { , , · · · , x } , ϵ ∈ { + , −} , s ∈ { p , t } n , o ∈ { p , t } and τ ∈ { p , t } M , we will use W iϵ ( τ , o , s ) or W iϵ ( τ , o ) to denote the web in (cid:18) L iϵ (cid:19) with indexing data ( τ , o , s ) or ( τ , o ) , as appropriate.Analogously, we write W ( τ ) and W (cid:48) ( τ ) for τ -indexed webs in (cid:110) L (cid:111) and (cid:110) L (cid:48) (cid:111) respectively. If the indexingdata is fixed, we will sometimes omit it from the notation (e.g. W i ± = W i ± ( τ , o , s ) and W = W ( τ ) ) and saythat the webs W i + and W i − correspond to each other.If f is a chain map and V and W are webs in the source and target complexes, then we write f ( V , W ) for the component of f from V to W . Lemma 3.9.
Suppose s ∈ { p , t } n , o ∈ { p , t } and τ ∈ { p , t } M . For ≤ i ≤ x − we have W i + ( τ , o , s ) = W i − ( τ , o , s ) as webs, and for i ∈ { , x } we have W i + ( τ , o ) = W i − ( τ , o ) as webs. Moreover, gr ext ( W i + ( τ , p , s )) = − gr ext ( W i − ( τ , p , s )) . Reidemeister I moves.
The Reidemeister I chain maps are the following. (cid:21)(cid:20) (cid:21) ∅ dcap cup (cid:20) (cid:21)(cid:20) (cid:21) ∅ cap dcup (cid:20) (cid:21)(cid:20) (cid:21) ∅ dcap cup (cid:20) (cid:21)(cid:20) (cid:21) ∅ cap dcup Here cap and cup simply denote the cap and cup foams, while dcap and dcup denote decorated capand cup foams. The decoration is by the polynomial (cid:205) a + b = N − X a Y b where X denotes the dot on thestrand and Y the dot on the circle, c. f. (2.2). Lemma 3.10.
The Reidemeister I chain maps R ± : (cid:110) L (cid:111) → (cid:18) L ± (cid:19) and R − ± : (cid:18) L x ± (cid:19) → (cid:110) L (cid:48) (cid:111) preserve theinternal and external homological degrees individually. Moreover, their only non-zero components are inexternal homological grading zero. Lemma 3.11.
In external homological grading zero, we have R − = p ◦ R + and R − + = R − − ◦ p (cid:48) where p and p (cid:48) are chain maps of decorated identity foams such that R − − ◦ R − ◦ · · · ◦ R − ◦ R − ◦ p = p (cid:48) ◦ R − − ◦ R − ◦ · · · ◦ R − ◦ R − Proof.
The chain maps p and p (cid:48) each consist of identity foams decorated by the polynomial (cid:205) a + b = N − X a Y b .In p , the dots X and Y are placed next to the Reidemeister I crossing, as shown inthe first picture on the right. These dots are spatially separated from the region inwhich the R R R R p (cid:48) . β ∗ ∗ → β ∗ ∗ (cid:3) Reidemeister II moves.
We will use Elias–Khovanov’s Soergel calculus [EK10a] to describe thechain maps associated to Reidemeister II and III moves. The Soergel calculus of type A n − is a graphicalincarnation of the 2-category of Soergel bimodules, which categorifies the Hecke algebra for S n . For any N ≥
2, it admits a 2-functor to the monoidal subcategory of
Foam N of webs and foams with 2 n boundarycomponents with suitable orientations. Instead of describing these 2-functors formally, we will just usethe Soergel calculus as shorthand notation for foams using the following dictionary: • In the A calculus, we have only a blue object, which we will interpret as the two strand web (cid:55)→ . • In the A calculus, we have red and blue objects, interpreted as three strand webs (cid:55)→ , (cid:55)→ . • Start dots and end dots (in any color) correspond to zip and unzip foams. • The trivalent vertices and correspond to digon creation and annihilation foams respectively.We also use cups : = ◦ and caps : = ◦ . • The 6-valent vertex corresponds to the foam shown in Figure 1.The Reidemeister II chain maps are the following. (cid:22) (cid:23)(cid:22) (cid:23) −∅ ∅ (cid:2) − (cid:3) (cid:20) (cid:21) (cid:22) (cid:23)(cid:22) (cid:23) −∅ ∅ (cid:2) (cid:3) (cid:20) − (cid:21) In both cases we have chosen to order the crossings from the top to the bottom. Now we can recordtwo observations.
Lemma 3.12.
The chain maps R ± : (cid:18) L ± (cid:19) → (cid:18) L ± (cid:19) and R − ± : (cid:18) L x − ± (cid:19) → (cid:18) L x ± (cid:19) preserve the internaland external homological grading individually and their only non-zero components involve palindromicresolutions. Lemma 3.13.
Let W ± = W ± ( τ , o ) and W x ± = W ± ( τ , o ) be pairs of corresponding webs in (cid:18) L ± (cid:19) and (cid:18) L x ± (cid:19) respectively. Further, let s ∈ { p , t } n be a palindrome in which t appears k times, and consider W ± = W ± ( τ , o , s ) and W x − ± = W x − ± ( τ , o , s ) in (cid:18) L ± (cid:19) and (cid:18) L x − ± (cid:19) respectively. Then we have R − ( W − , W − ) = (− ) k R + ( W + , W + ) R − + ( W x − + , W x + ) = (− ) k R − − ( W x − − , W x − ) . Proof.
In a single Reidemeister II move, the identity resolution is always sent to the identity resolution viathe identity. The maps involving the resolution with two thick edges are negatives of each other, whencomparing the two types of Reidemeister II moves with fixed order of crossings as in (3.2). (cid:3)
Reidemeister III moves.
In (3.1) we encounter four types of Reidemeister III moves. Namely, the moving strand can pass in front of or behind a positive or a negative crossing. In the following we showthe front and back versions alongside each other. In every case, the moving strand is the one connectingthe bottom left and top right boundary points.In each variant of Reidemeister III, we order the crossings in each tangle from top to bottom. The partsof the complexes with internal homological degree zero—where the internal crossing is resolved in theparallel fashion—are highlighted in blue. The parts with internal homological degree ± y ; shortly we shall specialize to y = (cid:24) (cid:25)(cid:24) (cid:25) − −−−− −−− (cid:24) (cid:25)(cid:24) (cid:25) −−−−−− − − - 0 - 0(y-1) -y
00 0 (1-y)0 y
Next, we consider the two ways in which the moving strand may pass a negative crossing:(3.4) (cid:24) (cid:25)(cid:24) (cid:25) − −−− −− − − - 00 00 (1-y) -y -0 (y-1) (cid:24) (cid:25)(cid:24) (cid:25) −−−− −−−− For the remainder of this paper we specialise to the choice y =
0. (Note in particular that the statementsimmediately below are not true for other choices!)
Lemma 3.14.
The chain maps R ± : (cid:18) L i ± (cid:19) → (cid:18) L i + ± (cid:19) in (3.1) do not decrease the external homologicalgrading.Proof. Since chain maps are of homological degree zero, the statement is equivalent to saying that theReidemeister III chain maps in (3.1) never increase the internal homological grading. This can be verifiedby inspecting (3.3) and (3.4). For the reader’s convenience we have highlighted the components of negativeinternal homological degree in green. All other non-zero components are highlighted blue or yellowand have internal homological degree zero because they map between the yellow and blue layers of therelevant complexes. Thus we only need to worry about components of the chain map which are not highlighted in the diagrams above. With y =
0, these components all vanish. (cid:3)
In other words, the Reidemeister III maps are filtered with respect to the filtration determined by theinternal homological degree, which we shall call the internal filtration . roposition 3.15. The filtration-preserving components of the chain maps R + : (cid:18) L i + (cid:19) → (cid:18) L i + + (cid:19) and R − : (cid:18) L i − (cid:19) → (cid:18) L i + − (cid:19) agree if ≤ i < x − . More precisely, we have R + ( W i + , W i + + ) = R − ( W i − , W i + − ) for pairs of corresponding webs W i ± in (cid:18) L i ± (cid:19) and W i + ± in (cid:18) L i + ± (cid:19) with gr ext ( W i ± ) = gr ext ( W i + ± ) .Proof. By inspecting (3.3) and (3.4) — for each of the 1+9+9+1 components of the R + chain map, checkthat the corresponding component of the R − chain map is the same (recalling y = (cid:3) Corollary 3.16.
The filtration-preserving component of the chain maps R + ◦ · · · ◦ R + : (cid:18) L + (cid:19) → (cid:18) L x − + (cid:19) and R − ◦ · · · ◦ R − : (cid:18) L − (cid:19) → (cid:18) L x − − (cid:19) agree. More precisely, we have R + ( W + , W x − + ) = R − ( W − , W x − − ) for pairs of corresponding webs W ± in (cid:18) L ± (cid:19) and W x − ± in (cid:18) L x − ± (cid:19) with gr ext ( W ± ) = gr ext ( W x − ± ) .Remark. The Reidemeister III chain maps shown in (3.3) and (3.4), their inverses, and four additionalvariations can be found in Elias–Krasner [EK10b]. Note, however, the following differences in conventions.Their positive crossings are our negative crossings and the crossings in their braids are ordered frombottom to top, while we order them from top to bottom. Finally, they read Soergel diagrams from left toright, while we read them from right to left.3.6.
Proof of the sweep-around property.
We can now assemble a proof of Theorem 3.3.
Proof of Theorem 3.3.
We need to show that the two chain maps sw + and sw − from (3.1) are equal. Forthis, let W and W (cid:48) be webs in (cid:110) L (cid:111) and (cid:110) L (cid:48) (cid:111) respectively. We shall compare the components of sw + andsw − between W and W (cid:48) .By Proposition 3.15, the R ± maps do not decrease the external homological degree, but by Lemmas 3.10and 3.12, the R ± ± and R ± ± maps preserve the external homological degree. Since gr ext ( W ) = gr ext ( W (cid:48) ) = R ± do not contribute to sw + or sw − . Now suppose that W ± are correspond-ing webs in (cid:18) L ± (cid:19) and W x − ± are corresponding webs in (cid:18) L x − ± (cid:19) with gr ext ( W ± ) = gr ext ( W x − ± ) =
0. Then,by Corollary 3.16, we have ( R + ◦ · · · ◦ R + )( W + , W x − + ) = ( R − ◦ · · · ◦ R − )( W − , W x − − ) . Let us also record that if R ± ◦ · · · ◦ R ± has a non-zero component between two webs W and W x − , thenfirst n digits of t ( W ) agree with the first n digits of t ( W x − ) . (Recall that the first n digits describe therightmost n crossings, which are spatially separated from the region in which Reidemeister III movesoccur.)Next we consider the pair of corresponding webs W ± = W ± ( s ( W ) , p ) in (cid:18) L ± (cid:19) , which appear in the imageof W under R ± , and the pair of corresponding webs W x ± = W x ± ( s ( W (cid:48) ) , p ) in (cid:18) L x ± (cid:19) , which have W (cid:48) as imageunder R − ± . The components of R − ± ◦ R ± ◦ · · · ◦ R ± ◦ R ± between these webs are sums over componentsthrough many possible intermediate webs W ± and W x − ± . By the previous argument, the ReidemeisterIII portions of the + - and the − -version of the map agree. By Lemma 3.13, the Reidemeister II portionscould at most cause a sign-discrepancy. However, since the first halves of t ( W ± ) , t ( W ± ) , . . . , t ( W x − ± ) allagree, and since Reidemeister II chain maps are zero on non-palindromic webs by Lemma 3.12, there isno sign-discrepancy. Thus, we record: ( R − + ◦ R + ◦ · · · ◦ R + ◦ R + )( W + , W x + ) = ( R − − ◦ R − ◦ · · · ◦ R − ◦ R − )( W − , W x − ) inally, we use Lemma 3.11 to compute:sw − ( W , W (cid:48) ) = ( R − − ◦ R − − ◦ R − ◦ · · · ◦ R − ◦ R − ◦ R − )( W , W (cid:48) ) = ( R − − ◦ R − − ◦ R − ◦ · · · ◦ R − ◦ R − ◦ p ◦ R + )( W , W (cid:48) ) = ( R − − ◦ p (cid:48) ◦ R − − ◦ R − ◦ · · · ◦ R − ◦ R − ◦ R + )( W , W (cid:48) ) = ( R − + ◦ R − − ◦ R − ◦ · · · ◦ R − ◦ R − ◦ R + )( W , W (cid:48) ) = ( R − + ◦ R − + ◦ R + ◦ · · · ◦ R + ◦ R + ◦ R + )( W , W (cid:48) ) = sw + ( W , W (cid:48) ) This completes the proof. (cid:3)
4. Khovanov–Rozansky homology in S From now on, we will only consider framed oriented links and framed oriented link cobordisms.Furthermore, all diffeomorphisms are oriented.4.1.
Link homology in abstract 3-balls.
The purpose of this section is to define a functorial Khovanov–Rozansky link homology for links in abstract 3-manifolds (abstractly) diffeomorphic to R , which isfunctorial under link cobordisms in abstract 4-manifolds diffeomorphic to R × [ , ] . The framework setup in this section could have been developed immediately after the initial construction of functorial linkinvariants, but to our knowledge it has not been developed in the literature. We hope that the carefulpresentation of this improvement of the invariant will be a helpful warm-up for the following section,where we employ a very similar strategy to build invariants of links in abstract 3-spheres. (cid:26) link embeddings in oriented B (cid:27) R link cobordisms in oriented W (cid:27) R × [ , ] up to isotopy rel ∂ (cid:27) KhR N −−−−→ (cid:26) bigraded vector spaceshomogeneous linear maps (cid:27) We will call such an invariant a link homology for links in -balls .Throughout this section, B will denote a : an oriented 3-manifold that is diffeomorphic to R via some (unspecified!) diffeomorphism. We say a link embedding L in R is generic if it is in genericposition with respect to the projection along the z -axis to R and all crossings in the resulting link diagramhave distinct y coordinates. In this case, we consider the crossings as ordered from smallest to largest y coordinate. We say a link embedding L in R is blackboard-framed if the framing is parallel to R . Lemma 4.1.
Let L ⊂ B be a link embedded in a 3-ball. Let ϕ and ϕ be two diffeomorphisms from B to R such that ϕ ( L ) and ϕ ( L ) are generic. Then we have the following: (1) There exists a continuous family of diffeomorphisms ϕ t for t ∈ [ , ] , such that ϕ t ( L ) is generic forall but finitely many t ∈ [ , ] , at which a Reidemeister move occurs or the crossing height orderchanges. (2) Given two such families ϕ t , and ϕ t , , both interpolating between ϕ and ϕ , then there exists acontinuous family ϕ t , s of diffeomorphisms interpolating between the families ϕ t , and ϕ t , , for whichthe parameter space [ , ] × [ , ] is stratified such that: • ϕ s , t ( L ) is generic for ( s , t ) in any codimesion-0 stratum, • ϕ s , t ( L ) undergoes a Reidemeister move or the crossing height order changes as ( s , t ) crossesthrough a codimension-1 stratum, • ϕ s , t ( L ) has a movie move as monodromy if ( s , t ) loops around a codimension-2 stratum.Proof. These facts follow from [CS93, CRS97]. (cid:3) efinition 4.2. Let B be an oriented 3-manifold diffeomorphic to R . We define M ( B ) def = { diffeomorphisms ϕ : B → R } . Given an embedded link L ⊂ B , we define the subspace M ( B , L ) def = { ϕ ∈ M ( B )| ϕ ( L ) is z -generic and blackboard-framed } , and consider the bundle π : T ( B , L ) → M ( B , L ) of bigraded vector spaces, whose fiber at the point ϕ isKhR N ( ϕ ( L )) .For a path ϕ t in M ( B ) between points ϕ , ϕ ∈ M ( B , L ) ⊂ M ( B ) , we define the grading-preservingisomorphism (cid:0) KhR N ( ϕ t ) : T ( B , L ) ϕ → T ( B , L ) ϕ (cid:1) def = ( KhR N ( ϕ t ( L )) : KhR N ( ϕ ( L )) → KhR N ( ϕ ( L ))) , where the latter denotes the linear map associated to the trace of the link isotopy ϕ t ( L ) in R × [ , ] .This is well-defined by Theorem 2.6, even though for some t the embeddings ϕ t ( L ) can be highly non-generic with respect to projection in the z -coordinate. Also note that while Reidemeister I moves induce q -grading shifts on the level of KhR N , any isotopy of framed links features such moves in pairs, leadingto a grading-preserving isomorphism. Lemma 4.3.
The parallel transport isomorphisms
KhR N ( ϕ t ) define a flat connection on T ( B , L ) .Proof. Lemma 4.1 (1) implies that we have such parallel transport maps KhR N ( ϕ t ) between the fibers overany pair of points ϕ and ϕ in the base. Lemma 4.1 (2) and Theorem 2.6 imply that the parallel transportmaps between the fibers do not depend on the choice of the path ϕ t . (cid:3) Definition 4.4.
Let L ⊂ B be a link embedded in a 3-ball. Then we define the Khovanov–Rozanskyhomology of L in B to be KhR N ( B , L ) def = Γ flat ( T ( B , L )) , the bigraded vector space of flat sections of the bundle T ( B , L ) .Note that every diffeomorphism ϕ : B → R such that ϕ ( L ) is generic and blackboard-framed inducesa grading-preserving isomorphism KhR N ( B , L ) → KhR N ( ϕ ( L )) by evaluating sections at the point ϕ . Definition 4.5.
Consider a link cobordism Σ ⊂ W in a 4-manifold W diffeomorphic to R × [ , ] . Let Σ in ⊂ W in and Σ out ⊂ W out denote the boundary links in the incoming and outgoing boundary 3-balls of W . Then we define KhR N ( W , Σ ) : KhR N ( W in , Σ in ) → KhR N ( W out , Σ out ) in two steps. First we pick a diffeomorphism ϕ : W → R × [ , ] , such that ϕ out : = ϕ | W out : W out → R and ϕ in : = ϕ | W in : W in → R are such that ϕ in ( Σ in ) and ϕ out ( Σ out ) are both generic and blackboard-framed.Then we declare KhR N ( W , Σ )( η ) , for a flat section η ∈ KhR N ( W in , Σ in ) , to be the unique flat section ofKhR N ( W out , Σ out ) with value: KhR N ( W , Σ )( η )( ϕ out ) = KhR N ( ϕ ( Σ ))( η ( ϕ in )) Lemma 4.6.
KhR N ( W , Σ ) is independent of the choices of ϕ in , ϕ out and ϕ , and thus well-defined.Proof. We first show independence of ϕ , given a fixed choice of ϕ in and ϕ out . Suppose that ϕ (cid:48) : W → R × [ , ] is another diffeomorphism restricting to ϕ in and ϕ out on W in and W out respectively.Lemma 4.7, proved below, implies that the link cobordisms ϕ ( Σ ) and ϕ (cid:48) ( Σ ) are isotopic rel boundaryin R × [ , ] and we have KhR N ( ϕ ( Σ )) = KhR N ( ϕ (cid:48) ( Σ )) by Theorem 2.6. ext we show independence of ϕ in , given a fixed choice of ϕ out . Let ϕ (cid:48) in : W in → R be anotherdiffeomorphism such that ϕ (cid:48) in ( Σ in ) is generic and blackboard-framed, and ϕ (cid:48) another diffeomorphism W → R × [ , ] restricting to ϕ (cid:48) in on W in but still to ϕ out on W out . Then, by Lemma 4.1 (1), we can find afamily ϕ in , t connecting ϕ in to ϕ (cid:48) in . By definition of parallel transport, we have: η ( ϕ (cid:48) in ) = KhR N ( ϕ in , t ( Σ in ))( η ( ϕ in )) Now we obtain a new diffeomorphism ϕ (cid:48) ◦ ϕ in , t : W → R × [ , ] and by the previous independenceresult and Theorem 2.6, we have:KhR N ( ϕ (cid:48) ( Σ ))( η ( ϕ (cid:48) in )) = KhR N ( ϕ (cid:48) ( Σ ) ◦ ϕ in , t ( Σ in ))( η ( ϕ in )) = KhR N ( ϕ ( Σ ))( η ( ϕ in )) Thus, the definition was independent of the choice of ϕ in . An analogous argument also establishesindependence of the choice of ϕ out . (cid:3) It remains to prove the lemma referenced above.
Lemma 4.7.
Let Σ ⊂ R ×[ , ] be a link cobordism and let f : R ×[ , ] → R ×[ , ] be a diffeomorphismwhich restricts to the identity in a neighborhood of the boundary R × { , } . Then Σ is isotopic rel boundaryto f ( Σ ) .Proof. The proof would be easy if we knew that f were isotopic to the identity, but π ( Diff + ( R ×[ , ] , R ×{ , })) is unknown. We can, however, replace f with a diffeomorphism f (cid:48) which is isotopic (rel boundary)to f , or replace f with a diffeomorphism f (cid:48) which coincides with f in a neighborhood of Σ . In both cases,proving that f (cid:48) ( Σ ) is isotopic to Σ easily implies that f ( Σ ) is isotopic to Σ .Choose a point p ∈ R such that p × [ , ] is disjoint from Σ . There is no obstruction to modifying(post-composing) f by an isotopy which takes f ( p × [ , ]) to p × [ , ] , so we may assume that f restrictsto the identity on p × [ , ] .Next consider the tangent map of f along p × [ , ] . We would like to deform the tangent mapto the identity, but there is an obstruction living in π ( SO ( )) (cid:27) Z /
2. We can modify (precompose) f in a neighborhood of p × [ , ] (and away from Σ ) so that this obstruction vanishes. (Specifically, let f : [ , ] → [ , ] be a smooth function such that f ( t ) = t near 0 and f ( t ) = t near 1. Let γ : [ , ] → SO ( ) be a representative of the nontrivial element of π ( SO ( )) , with γ ( ) = γ ( ) = .Let B be the unit ball in R , and for p ∈ B , let | p | denote the distance from p to the origin. Define adiffeomorphism of [ , ] × B by ( s , p ) (cid:55)→ ( s , γ ( f ( s · ( − | p |)))( p )) . This diffeomorphism is the identity near [ , ] × ∂ B and it effects a full twist on the tangent space along [ , ] × { } .)Once the above obstruction vanishes we can isotope f to a map which is the identity on a neighborhood N of p × [ , ] ∪ R × { , } .Choose a family of diffeomorphisms д t : R × [ , ] → R × [ , ] , with t ∈ [ , ] , such that д isthe identity, д t restricted to R × { , } is the identity for all t , and д ( Σ ) ⊂ N . The family of surfaces f ( д t ( Σ )) provides an isotopy from f ( Σ ) = f ( д ( Σ )) to f ( д ( Σ )) . But f is the identity on N and д ( Σ ) ⊂ N ,so f ( д ( Σ )) = д ( Σ ) . The family of surfaces д t ( Σ ) provides an isotopy from д ( Σ ) to д ( Σ ) = Σ . Composingthese two isotopies provides the desired isotopy from f ( Σ ) to Σ . (cid:3) Link homology in abstract 3-spheres.Definition 4.8. A link homology for links in -spheres is a functor (cid:26) link embeddings in oriented S (cid:27) S link cobordisms in oriented Y (cid:27) S × [ , ] up to isotopy rel ∂ (cid:27) −→ (cid:26) bigraded vector spaceshomogeneous linear maps (cid:27) heorem 4.9. KhR N extends to a link homology theory for links in -spheres. The proof occupies the remainder of this subsection.
Remark.
In the proof of Theorem 4.9, we will show that the sweep-around property from Theorem 1.1 issufficient to extend a link homology for links in 3-balls to 3-spheres, without using any special propertiesof KhR N . Definition 4.10.
Let S be an oriented 3-manifold diffeomorphic to S . For any point p ∈ S \ L , we consider L as a link in the 3-ball S \ { p } and denote by π : T ( S , L ) → S \ L the bundle of bigraded vector spaces,whose fiber at the point p ∈ S \ L is KhR N ( S \ { p } , L ) as defined in Definition 4.4.For any path p t in S \ L , we have that L × [ , ] ⊂ S × [ , ] \ {( p t , t )} t ∈[ , ] (cid:27) R × [ , ] . By the resultsof §4.1, this cobordism induces a parallel transport isomorphism ( KhR N ( p t ) : T ( S , L ) p → T ( S , L ) p ) def = KhR N ( S × [ , ] \ {( p t , t )} t ∈[ , ] , L × [ , ]) Lemma 4.11.
The parallel transport isomorphisms
KhR N ( p t ) define a flat connection on T ( S , L ) .Proof. We have to show that the parallel transport isomorphisms associated to closed loops p t in S \ L areidentity maps. Suppose first that p t is a contractible loop. Then the pair ( S ×[ , ]\{( p t , t )} t ∈[ , ] , L ×[ , ]) isdiffeomorphic to a pair ( R × [ , ] , Σ ) where Σ is isotopic to an identity link cobordism, which implies thatthe parallel transport isomorphism KhR N ( p t ) is the identity. This also implies that the parallel transportisomorphisms associated to isotopic paths between two points p and p in S \ L are equal. Now suppose thatthe loop p t is a small meridian around a component of L . Then the pair ( S × [ , ] \ {( p t , t )} t ∈[ , ] , L × [ , ]) is diffeomorphic to a pair ( R × [ , ] , Σ ) where Σ is a sweep-around cobordism as in (1.1). By Theorem 1.1,it follows that the parallel transport isomorphism KhR N ( p t ) is the identity. Since π ( S \ L ) is generatedby such small meridian loops, it follows that the parallel transport isomorphism for every loop p t is theidentity. (cid:3) Definition 4.12.
Let L ⊂ S be a link embedded in a 3-sphere. The we define the Khovanov–Rozanskyhomology of L in S to be KhR N ( S , L ) def = Γ flat ( T ( S , L )) , the bigraded vector space of flat sections of the bundle T ( S , L ) .Note that every point p ∈ S \ L induces a grading-preserving isomorphism KhR N ( S , L ) → KhR N ( S \{ p } , L ) of evaluating sections at the point p . Definition 4.13.
Consider a link cobordism Σ ⊂ W in a 4-manifold W diffeomorphic to S × [ , ] . Let Σ in ⊂ W in and Σ out ⊂ W out denote the boundary links in the incoming and outgoing boundary 3-spheresof W . Now we define KhR N ( W , Σ ) : KhR N ( W in , Σ in ) → KhR N ( W out , Σ out ) by first choosing a path p t ⊂ W \ Σ from p ∈ W in \ Σ in to p ∈ W out \ Σ out . Then we have W \ {( p t , t )} t ∈[ , ] (cid:27) R × [ , ] and we declare KhR N ( W , Σ )( η ) , for a flat section η ∈ KhR N ( W in , Σ in ) , to be the unique flatsection of KhR N ( W out , Σ out ) with valueKhR N ( W , Σ )( η )( p out ) = KhR N ( W \ {( p t , t )} t ∈[ , ] , Σ )( η ( p in )) Lemma 4.14.
KhR N ( W , Σ ) is independent of the choice of the path p t . roof. Let us first fix a choice of endpoints p ∈ W in \ Σ in and p ∈ W out \ Σ out . Then any two choices ofpaths p t ∈ and p (cid:48) t from p to p can be related by isotopy in W \ Σ or splicing in a little loop linking acomponent of Σ . As before, isotopic paths give rise to isotopic surfaces in R × [ , ] , which induce equalmaps. Similarly, in the case of a linking loop, we can choose a standard local model and then notice thatthe sweep-around property from Theorem 1.1 implies that the two paths induce the same map. Finally,the independence from the choice of endpoints p ∈ W in \ Σ in and p ∈ W out \ Σ out follows as in the proofof Lemma 4.6. (cid:3) This completes the proof of Theorem 4.9.4.3.
Monoidality.
Links in 3-balls and their cobordisms form a symmetric monoidal category underboundary connect sum, which is respected by KhR N as we will now see. Proposition 4.15.
The Khovanov–Rozansky homologies
KhR N are symmetric monoidal functors.Proof. Let L ∈ B and L ∈ B and write L for the resulting split disjoint union in B def = B ∂ B . Wecan find a diffeomorphism ϕ : B → R such that L is not only generic and blackboard-framed, but alsosuch that the z -projections of the L and L components of L are contained in disjoint disks in R . Then,monoidality is manifest in the definition of KhR N , and we getKhR N ( B ∂ B , L (cid:116) L ) (cid:27) KhR N ( ϕ ( L (cid:116) L )) = KhR N ( ϕ ( L )) ⊗ KhR N ( ϕ ( L )) (cid:27) KhR N ( B , L ) ⊗ KhR N ( B , L ) . The compatibility on the level of morphisms is verified similarly. (cid:3)
Given a finite collection of links in 3-balls L i ⊂ B i , we can also defineKhR N ((cid:116) B i , (cid:116) L i ) def = (cid:204) KhR N ( B i , L i ) . Then the proof of the proposition implies that the boundary connect sum of 3-balls induces naturalisomorphisms KhR N ((cid:116) i B i , (cid:116) i L i ) (cid:27) KhR N ( ∂ B i , (cid:116) i L i ) . Remark.
This monoidality property can be interpreted as saying that KhR N categorifies the gl N skeinalgebra of R . For more on skein algebra categorification we refer to [QW18b].5. A TQFT in dimensions 4 + ϵ In this and the following section we construct three alternative 4-categorical structures from Khovanov–Rozansky homology. (The three alternatives are not essentially different; they ought to be differentdescriptions of the same thing.) These are: • a “lasagna algebra”, which is a higher dimensional analogue of a planar algebra, introduced here, • a “disklike 4-category”, as defined in [MW12], • a “braided monoidal 2-category”, in the sense of [BN96].In fact, we use the construction of a lasagna algebra as a shortcut towards building a disklike 4-category.The construction of a braided monoidal 2-category is independent, and can be read separately. Theadvantage of the lasagna algebra and disklike 4-category approaches is that they immediately provideinvariants of oriented smooth 4-manifolds, valued in bigraded vector spaces. In this paper, we brieflydescribe these invariants but do not explore them further. n §6, we recast Khovanov–Rozansky homology in the more traditional framework of a braidedmonoidal 2-category KhR N with duals. We conjecture that the sweep-around property implies that thisbraided monoidal 2-category is an SO ( ) fixed point in the sense of Lurie [Lur09], and consequentlyleads to invariants of oriented 4-manifolds using the framework of factorization homology (see also[BZBJ18, BJS18] for related constructions one dimension down). We do not pursue this, preferring themore direct approach to oriented 4-manifold invariants described in this section.As before, links and link cobordisms are assumed to be oriented and framed, and all diffeomorphismsare oriented in this section.5.1. An algebra for the lasagna operad.
Throughout this section we assume familiarity with planaralgebras [Jon99]. L ⊂ S L ⊂ S L ⊂ S L ⊂ S Σ ΣΣ
Figure 2. A lasagna diagram, projected into 3d
Definition 5.1. A lasagna algebra L consists of • for each link L in a 3-sphere S , a (bigraded) vector space L( S , L ) , which depends functorially onthe pair ( S , L ) , • for each lasagna diagram D , which, by definition, consists of a 4-ball B , with a finite collection ofdisjoint 4-balls B i removed from the interior, with boundary components S (on the outside) and S i (the boundaries of the removed interior balls B i ), and properly embedded framed oriented surface Σ in the complementary region, meeting the boundary spheres in links L and L i (see Figure 2), a(homogeneous) linear map L( D ) : (cid:204) i L( S i , L i ) → L( S , L ) , such that • surfaces Σ and Σ (cid:48) which are isotopic rel boundary induce identical linear maps, • if f : D → D (cid:48) is a diffeomorphism between lasagna diagrams, then the square (cid:203) i L( S i , L i ) L( S , L ) (cid:203) i L( f ( S i ) , f ( L i )) L( f ( S ) , f ( L )) L( D ) (cid:203) i L( f | Si ) L( f | S )L( D (cid:48) ) ommutes, • a ‘radial’ surface L × [ , ] ⊂ S × [ , ] induces the identity map L( S , L ) → L( S , L ) (or more precisly,mapping cylinders of diffeomorphisms induce the same map specified for that diffeomorphism byfunctoriality), • and gluing of a ‘smaller’ lasagna diagram into one of the removed balls of a ‘larger’ lasagnadiagram (with compatible boundaries) to obtain a single lasagna diagram is compatible with thecorresponding composition of linear maps.We won’t actually spell this out in detail, but one can easily extract from this definition the notion ofthe lasagna operad (actually a coloured operad, with colours corresponding to links), and that a lasagnaalgebra is an algebra for that operad. One can of course consider lasagna algebras valued in symmetricmonoidal categories other than (bigraded) vector spaces.The ‘one input ball’ part of a lasagna algebra is essentially equivalent to a functorial invariant of linksin 3-spheres: we have a vector space for each such link, and linear maps for cobordisms between them,which compose appropriately. It is not immediately clear that any such functorial invariant extends toa full lasagna algebra, with well-defined operations for multiple input balls. The goal in this section isto show that this is the case for Khovanov–Rozansky homology. In fact, our argument shows that anyfunctorial invariant of links and cobordisms in 3-spheres which satisfies the monoidality property andsweep-around move extends to a lasagna algebra. Theorem 5.2.
Khovanov–Rozansky homology affords the structure of a lasagna algebra.Proof.
For a lasagna diagram D (as in Figure 2) we define a linear mapKhR N ( D ) : (cid:204) i KhR N ( S i , L i ) → KhR N ( S , L ) , as follows. We first choose points q j ∈ S j \ L j and q ∈ S \ L and then a properly embedded 1-complex T ⊂ B , disjoint from Σ , such that the underlying graph of T is a tree and the end points of the 1-complexare { q j , q } . Choose a small closed tubular neighborhood N of T , also disjoint from Y . The complement of N in B \ (cid:116) B i is diffeomorphic to R × [ , ] with some embedded surface Σ (cid:48) . We will view Σ (cid:48) as a bordismbetween two links in two copies of R . One copy is identified with S q : = S \ { q } , which contains the link L . The other copy X is the remainder of the boundary, and can be expressed as the boundary connectsum of the 3-balls S q i i : = S i \ { q i } , connected along the tree T . The 3-ball X contains the distant union ofthe links L i . Khovanov–Rozansky homology for links in 3-balls gives us a mapKhR N ( Σ (cid:48) ) : KhR N ( X , (cid:116) i L i ) → KhR N ( S q , L ) which, together with the monoidality isomorphism from §4.3, specifies a mapKhR N ( D ) : (cid:204) i KhR N ( S i , L i ) (cid:27) (cid:204) i KhR N ( S q i i , L i ) T (cid:27) KhR N ( X , (cid:116) i L i ) → KhR N ( S q , L ) (cid:27) KhR N ( S , L ) . Here the first and last maps are the ‘evaluation’ isomorphisms discussed below Definition 4.12, and wehighlight that the monoidality isomorphism depends on the tree T .We must check that the overall map above does not depend on the choices of q i and T . This isstraightforward, so we merely sketch the argument. Isotoping the points q i does not change the map,by the same argument that showed that KhR N is well-defined for links in 3-spheres; see §4.2. Isotoping T disjointly from Σ clearly does not affect the map. Changing the combinatorics of the underlying treeof T can be done in such a way that N varies continuously and remains far from Σ , and so does notaffect the map. Isotoping T through Σ does not affect the map, thanks to the sweep-around property (seeTheorem 1.1 above). Thus KhR N ( Σ ) is well-defined. ext we must show compatibility with the operad composition. For this we consider three lasagnadiagrams: • D with output boundary ( S , L , q ) , with input boundaries ( S i , L i , q i ) i ∈ J , surface Σ , and tree T , • D with output boundary ( S , L , q ) , with input boundaries ( S , L , q ) along with ( S i , L i , q i ) i ∈ K ,surface Σ , and tree T • D , the result of gluing D inside D , with outer boundary ( S , L , q ) , input boundary ( S i , L i , q i ) i ∈ J ∪ K ,surface Σ = Σ ∪ Σ , and tree T = T ∪ T .Compatibility with the operad composition now boils down to the following claim:KhR N ( D ) = KhR N ( D ) ◦ ( KhR N ( D ) ⊗ ) : (cid:32)(cid:204) i ∈ J KhR N ( S i , L i ) (cid:33) ⊗ (cid:32)(cid:204) i ∈ K KhR N ( S i , L i ) (cid:33) → KhR N ( S , L ) We compare these two linear maps on the level of 3-ball link homologies, that is, with respect to a fixedchoice of base points q i and q , and we suppress associators. On this level KhR N ( D ) is determined by thelinear map(5.1) (cid:32)(cid:204) i ∈ J KhR N ( S q i i , L i ) (cid:33) ⊗ (cid:32)(cid:204) i ∈ K KhR N ( S q i i , L i ) (cid:33) T (cid:27) KhR N ( X , (cid:116) i L i ) → KhR N ( S q , L ) where X denotes the boundary connect sum of the 3-balls S q i i for i ∈ J ∪ K along the tree T , and the firstisomorphism is provided by monoidality. On the other hand, the linear map KhR N ( D ) ◦ ( KhR N ( D ) ⊗ ) is determined by the composite (cid:32)(cid:204) i ∈ J KhR N ( S q i i , L i ) (cid:33) ⊗ (cid:32)(cid:204) i ∈ K KhR N ( S q i i , L i ) (cid:33) T ⊗ (cid:27) KhR N ( X J , (cid:116) i ∈ J L i ) ⊗ (cid:32)(cid:204) i ∈ K KhR N ( S q i i , L i ) (cid:33) KhR N ( Σ (cid:48) )⊗ −−−−−−−−−→ KhR N ( S q , L ) ⊗ (cid:32)(cid:204) i ∈ K KhR N ( S q i i , L i ) (cid:33) T (cid:27) KhR N ( X K , (cid:116) i ∈{ }∪ K L i ) KhR N ( Σ (cid:48) ) −−−−−−−→ KhR N ( S q , L ) . Here we write X J for the boundary connect sum of the 3-balls S q i i : = S i \ { q i } for i ∈ J that is determinedby T , and X K for the boundary connect sum of the S q i i for i ∈ { } ∪ K along T . After commuting themap induced by Σ (cid:48) past the second monoidality isomorphism, we arrive at(5.2) (cid:32)(cid:204) i ∈ J KhR N ( S q i i , L i ) (cid:33) ⊗ (cid:32)(cid:204) i ∈ K KhR N ( S q i i , L i ) (cid:33) T (cid:27) KhR N ( X , (cid:116) i ∈ J ∪ K L i ) KhR N ( Σ (cid:48) ◦( Σ (cid:48) ∪ )) −−−−−−−−−−−−−→ KhR N ( S q , L ) . Since the link cobordism Σ (cid:48) ◦ ( Σ (cid:48) ∪ ) is isotopic to Σ , the functoriality of KhR N implies that the maps in(5.1) and (5.2) are equal. This proves the claim. (cid:3) Skein theory for lasagna algebras.
In this section, we use the lasagna algebra described aboveto construct an invariant S N ( W ; L ) of smooth oriented 4-manifolds W , possibly with a link L in theboundary, valued in bigraded vector spaces. It is akin to the skein modules of 3-manifolds, which can bedefined from any ribbon category, except that everything happens one dimension higher. The relationshipbetween this invariant and what we are eventually after is analogous to that between H H and H H ∗ of analgebra. efinition 5.3. Let W be a smooth oriented 4-manifold and L ⊂ ∂ W a link. A lasagna filling F of W with boundary L consists of the following data • a finite collection of ‘small’ 4-balls B i embedded in the interior of W ; • an framed oriented surface Σ properly embedded in X \ (cid:116) i B i , meeting ∂ W in L and meeting each ∂ B i in a link L i ; and • for each i , a homogeneous label v i ∈ KhR N ( ∂ B i , L i ) .The bidegree of F is deg ( F ) : = (cid:205) i deg ( v i ) + ( , ( − N ) χ ( Σ )) . We will also consider linear combinationsof lasagna fillings and impose the relation that lasagna fillings are multilinear in the input labels v i . Thus,lasagna fillings of W with boundary L form a bigraded vector space.For a 4-ball W with a link L ∈ ∂ W , a lasagna filling F is equivalent to the data of a lasagna diagram D together with input labels v i ∈ KhR N ( S i , L i ) . In particular, we can compute the evaluation KhR N ( F ) = KhR N ( D )( v i ) ∈ KhR N ( ∂ W , L ) . Definition 5.4.
Let W be a smooth oriented 4-manifold and L ⊂ ∂ W a link. Then we define the bigradedvector space S N ( W ; L ) def = Z { lasagna fillings F of W with boundary L }/∼ where ∼ is the transitive and linear closure of the relation on lasagna fillings for which F ∼ F if F hasan input ball B with label v , and F can be obtained from F by replacing B with a third lasagna filling F of a 4-ball such that v = KhR N ( F ) , followed by an isotopy rel boundary. This is illustrated in Figure 3. F v v k ∼ F F v i v j v k Figure 3.The relation ∼ is homogeneous and, thus, S N ( W ; L ) is a bigraded vector space since the bidegree ofa cobordism map KhR N ( Σ ) is ( , ( − N ) χ ( Σ )) and the Euler characteristic of surfaces is additive undergluing along links. Remark . If L represents a non-zero class in H ( W ) , then we have S N ( W ; L ) = ∅ since there are nocompatible lasagna fillings. It would be interesting to relate S N to gl N versions of the invariants of linksin S × S or ( S × S ) д constructed by Rozansky [Roz10] and Willis [Wil18] respectively, which can benon-trivial for homologically non-zero links. Example . If W is a standard 4-ball with L ⊂ ∂ W = S , then the evaluation of lasagna fillings inducesan isomorphism ev : S N ( W ; L ) (cid:27) KhR N ( S , L ) . In other words, the above complicated quotient yields theusual KhR N ( S , L ) in this case. roof. It follows from Theorem 5.2 that equivalent lasagna fillings of W have equal evaluation. Thus, weget a well-defined linear map ev : S N ( W ; L ) → KhR N ( S , L ) , which is surjective since any homogeneous v ∈ KhR N ( S , L ) appears as the image of a radial lasagna filling F v . Similarly, if two lasagna fillings F and F have equal evaluation v ∈ KhR N ( S , L ) , then we observe F ∼ F v ∼ F , and so ev is injective. (cid:3) Having defined the skein module S N ( W ; L ) , we now proceed to constructing a disklike 4-category.This will also lead to a more refined invariant, taking the form of a chain complex with 0-th homology S N ( W ; L ) .5.3. A disklike 4-category.
We very briefly recall the key points of the definition of a disklike 4-category,from §6 of [MW12]. A disklike n -category C consists of: • for each 0 ≤ k ≤ n , a functor C k : { k -balls and diffeomorphisms } → Set (and we interpret C k ( X ) as the set of k -morphisms with shape X ), • for each k − Y in the boundary of a k -ball X , a restriction map C k ( X ) → C k − ( Y ) (to be morecareful, these restriction maps only need to be defined on sufficiently large subsets of C k ( X ) , forexample to allow for transversality issues), • for each k -ball X presented as the gluing of two k -balls X and X along a common k − Y intheir boundaries, a gluing map C k ( X ) × C k − ( Y ) C k ( X ) → C k ( X ) , • such that these gluing operations are compatible with the action of diffeomorphisms, and associa-tive on the nose, • and that two diffeomorphisms of n -balls which are isotopic rel boundary act identically, • along with some data and axioms concerning identities which we omit here.(As a reminder, the surprising feature of this definition is that while gluing is required to be strictlyassociative, this definition actually models fully weak n -categories. The key point is that we do not choosecanonical models for the shape of a k -morphism, and it is up to ‘the end user’ to pick reparametrisationsof glued balls back to any standard model balls that they prefer. It is these reparametrisations that areresponsible for introducing all the difficult structural isomorphisms of most definitions. This is analogousto the idea of a Moore loop space, which has a strictly associative composition, versus an ordinary loopspace, which has a complicated higher associator structure described by Stasheff polyhedra.)As explained in [MW12], one of the primary examples of a disklike n -category is string diagrams fora pivotal traditional n -category. This string diagram construction works just as well for a lasagna algebra(which is essentially a pivotal 4-category with trivial 0- and 1-morphisms). Specifically, starting from thelasagna algebra KhR N , we define a disklike 4-category KhR N as follows: • For X a 0-ball, we define KhR N ( X ) to be a single-element set. • For X a 1-ball, we define KhR N ( X ) to be a single-element set. • For X a 2-ball, we define KhR N ( X ) to be the set of all configurations of finitely many framedoriented points in X . • For X a 3-ball, we define KhR N ( X ) to be the set of all framed oriented tangles ( not up to isotopy)properly embedded in X . If c is a finite configuration of oriented points in ∂ X , we define KhR N ( X ; c ) to be the set of all oriented tangles which restrict to c on ∂ X . • For X a 4-ball, and L a link in ∂ X , we define KhR N ( X ; L ) to be the bigraded vector space S N ( X ; L ) defined above, that is, all lasagna fillings of X which restrict to L on the boundary, modulo relationsdescribed above. Recall that by Example 5.6 we know KhR N ( X ; L ) (cid:27) KhR N ( ∂ X , L ) . e define KhR N ( X ; L ) to be lasagna fillings modulo relations rather than simply defining it to beKhR N ( ∂ X , L ) in order to make it easier to define composition below.We will henceforth drop superscripts and write KhR N ( X ) instead of KhR kN ( X ) .In dimensions 0 through 3, it is clear that KhR N ( X ) is functorial with respect to diffeomorphisms.In dimension 4, it is clear the diffeomorphisms act on lasagna fillings; what remains is to show that therelations we impose are compatible with the action of diffeomorphisms. Specifically, for a diffeomorphism f we must show that if KhR N ( F ) = KhR N ( F (cid:48) ) then KhR N ( f ( F )) = KhR N ( f ( F (cid:48) )) . This follow from the factthat any diffeomorphism of a 4-ball (rel boundary) is isotopic to the identity away from a small 4-ball inthe interior. We can arrange that this small 4-ball is disjoint from Σ and the B i . The argument is similar to(but simpler than) the argument given in Lemma 4.7.We must now define gluing (composition) of morphisms. In dimensions 0 through 3 the morphismsare purely geometric and the gluing is defined to be the obvious geometric gluing of submanifolds. Indimension 4, there is again an obvious geometric gluing map of lasagna fillings. We must show thatthis gluing map is compatible with the relations we impose on fillings. This follows from the operadcomposition property proved in the previous section.Finally, the (omitted above) axioms about identities require that we check that 4-ball diffeomorphismswhich are supported away from the surface Σ act trivially. The diffeomorphism action on lasagna fillingsis just moving submanifolds around (and, if the internal balls move, applying the S -functoriality actionfrom the first piece of data for a lasagna algebra to the labels), so a diffeomorphism supported away fromthe surface and the internal balls does not change a lasagna filling.5.4. Blob homology.
Having built a disklike 4-category we immediately obtain an alternative descriptionof the skein module S N ( W ; L ) for a link in the boundary of any oriented smooth 4-manifold W , as firstintroduced in §5.2.This is the construction from [MW12, §6.3], which describes S N ( W ; L ) as a colimit, taken over allways of decomposing a 4-manifold W into a gluing of closed balls (with some regularity conditions onthe ways these balls meet). For any such decomposition, we draw compatible links in the boundariesof each of the balls (i.e. if two balls meet along some 3-manifold, the intersections of the two links withthat 3-manifold are tangles, and identical, and a similar condition for any ball meeting ∂ W ). Then thebigraded vector space at such a decomposition is the direct sum, over the choices of link labels, of thetensor products of the Khovanov–Rozansky homologies of each link. The arrows in the colimit diagramare ways of coarsening the decomposition by gluing several balls together into a single ball. The gluingmaps for a disklike 4-category provide morphisms of bigraded vector spaces. Finally, the skein moduleinvariant KhR N ( W ; L ) associated to W is just the colimit of this diagram.We will leave it as an exercise to the interested reader to verify that these two constructions actuallygive the same result!Our motivation for introducing the disklike 4-category is that the construction of [MW12, §6.3]actually gives much more. Associated to any link L in the boundary of a 4-manifold W , we obtain theblob complex (with coefficients in the disklike 4-category KhR N ), which we write as B ∗ ( KhR N )( W ; L ) .(One approach to the definition of this complex is by replacing the colimit described above with anappropriate homotopy colimit, cf [MW12, §7].) This has a new homological grading, unrelated to theinternal homological grading from Khovanov-Rozansky homology. The 0-th homology of this complexrecovers the bigraded vector space S N ( W ; L ) , but the higher blob homology groups, denoted S Ni ( W ; L ) for i >
0, potentially carry further information.Attempting any calculations of this invariant, or of its 0-th homology in either formulation, remainsbeyond the scope of the present paper, and developing appropriate computational tools is an open problem or future work. One such tool should come from a categorification of the gl N skein relation, namely theskein exact triangle for Khovanov–Rozansky chain complexes in R , which induces a long exact sequenceon homology groups. For a skein triple of links in the boundary of some interesting 4-manifold W wehave every reason to expect that the corresponding sequence on the level of the skein module S N is nolonger exact. We do, however, obtain long exact sequences on the level of the blob complex, which giverise to a spectral sequence that relates the skein modules S N for the three links. In fact, the study ofthese spectral sequences was the original motivation for the blob complex (however ahistorical this mightseem, given the publication dates).6. A pivotal braided monoidal dg 2-categoryIn this section, we define a semistrict braided monoidal 2-category KhR N in the sense of [KV94] (andin fact, in the stricter sense of [BN96]) from Khovanov–Rozansky homology. The spaces of 2-morphismsform bigraded vector spaces in the sense of Remark 2.8, and so we will sometimes add the adjective ‘dg’.Recall that one expects that braided monoidal 2-categories should be the same as 4-categories whichare ‘boring at the bottom two levels’, so there is a shift by two in the dimensions of the morphisms relativeto the previous section.The available definition of a braided monoidal 2-category has already been strictified quite a bit, andthis necessitates jumping through some hoops to even get started. Rather than defining the morphisms ofthe 2-category (which would be the 3-morphisms of the corresponding 4-category) to simply be arbitraryembedded tangles, we will need to introduce a particular combinatorial model of a tangle diagram. Definition 6.1.
The category TD of oriented tangle diagrams has objects given by finite words inthe alphabet {↑ , ↓} , including the empty word. The morphisms are admissible words in the alphabet { cup i , cap i , crossing i , crossing − i } i ≥ of generating morphisms.The realisation r ( t ) of a morphism t : A → B is a tangle diagram drawn in the square [ , ] × [ , ] byfirst placing the words A and B as collections of oriented tangle endpoints on [ , ] × { } and [ , ] × { } respectively, and then constructing an oriented tangle diagram starting from the bottom A by attachingcups, caps, crossings or inverse crossings with i parallel strands to the left, as specified by the t . Theword t is defined to be admissible if this procedure succeeds in generating an oriented tangle diagram.We will consider these diagrams up to individually rescaling the x - and y -coordinates in [ , ] × [ , ] byorientation-preserving diffeomorphisms of [ , ] . As such, every morphism in TD has a unique realisation,and we say that the morphism is the Morse data of the oriented tangle diagram.The composition of morphisms in TD is given by concatenating lists of generating morphisms.The remainder of this section contains the definition of the semistrict braided monoidal dg 2-category KhR N . We will first define this as a dg 2-category and subsequently add a semistrict monoidal structureand a braiding.6.1. A strict 2-category.
The strict dg 2-category
KhR N consists of the following data: • The objects are given by finite words in the alphabet {↑ , ↓} , including the empty word. • The 1-morphisms are admissible words in the alphabet { cup i , cap i , crossing i , crossing − i } i ≥ ofgenerating morphisms.The horizontal composition of 1-morphisms is given by concatenation of words, which is strictlyassociative. Given a pair of 1-morphisms f , д : A → B , the bigraded vector space of 2-morphisms from f to д is defined to be KhR N ( f , д ) : = KhR N ( Tr ( r ( f ) , r ( д ))) : = KhR N (cid:169)(cid:173)(cid:171) fд (cid:170)(cid:174)(cid:172) . Here the link diagram Tr ( r ( f ) , r ( д ) is constructed from the realisations r ( f ) and r ( д ) by reflecting r ( д ) in a horizontal line, reversing its orientations, composing with r ( f ) and closing off as shownin the figure . Note that this is well-defined because the Khovanov–Rozansky invariants of twolink diagrams, which are planar-isotopic through link diagrams with identical Morse data, arecanonically isomorphic.For 1-morphisms f , д : A → B and k , l : B → C , the horizontal composition of 2-morphisms KhR N ( f , д ) ⊗ KhR N ( k , l ) → KhR N ( f k , дl ) is defined as the homogeneous linear map computedas follows:KhR N (cid:169)(cid:173)(cid:171) fд (cid:170)(cid:174)(cid:172) ⊗ KhR N (cid:169)(cid:173)(cid:171) kl (cid:170)(cid:174)(cid:172) (cid:27) KhR N (cid:169)(cid:173)(cid:171) kl fд (cid:170)(cid:174)(cid:172) → KhR N (cid:169)(cid:173)(cid:171) kl fд (cid:170)(cid:174)(cid:172) → KhR N (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) fklд (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) Here we have used the canonical isomorphism between the tensor product of Khovanov–Rozansky homologies of two link diagrams and the homology of the split disjoint union of thediagrams, and then cobordism maps induced by a collection of saddles and a particular type ofplanar isotopy. Using functoriality of KhR N , it is easy to check that the horizontal composition isassociative.Now, for 1-morphisms f , д , h : A → B , the vertical composition of 2-morphisms KhR N ( f , д ) ⊗ KhR N ( д , h ) → KhR N ( f , h ) is defined as the homogeneous linear map computed as follows:KhR N (cid:169)(cid:173)(cid:171) fд (cid:170)(cid:174)(cid:172) ⊗ KhR N (cid:169)(cid:173)(cid:171) дh (cid:170)(cid:174)(cid:172) (cid:27) KhR N (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) дhfд (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) → KhR N (cid:169)(cid:173)(cid:171) fh (cid:170)(cid:174)(cid:172) Here, the interesting map is induced by a link cobordism which is cylindrical over the top andbottom quarters of the link diagrams, and which can be constructed as r ( д ) × halfcircle in themiddle. More explicitly, it consists of a composition of elementary cobordisms which cancel cupswith caps and positive with negative crossings in r ( д ) and its reflection.The identity 2-morphism f on a 1-morphism f : A → B is defined to be the image of the unitunder the linear map Z = KhR N (∅) → KhR N (cid:169)(cid:173)(cid:171) A A (cid:170)(cid:174)(cid:172) → KhR N (cid:169)(cid:173)(cid:171) ff (cid:170)(cid:174)(cid:172) which is induced by the link cobordism which first creates a collection of concentric circles asspecified by A , and then pairs of cups and caps, crossings and inverse crossings, to form r ( f ) composed with its reflection. It is a consequence of functoriality that the vertical composition We omit to indicate realisations r (−) in this and all following figures. f 2-morphisms is strictly associative and that f is indeed an identity 2-morphism. Finally, asimilar check establishes the interchange law that specifies the compatibility of the horizontal andvertical composition of 2-morphisms.6.2. A semistrict monoidal 2-category.
Next, we show that th 2-category
KhR N admits a semistrictmonoidal structure. Following [BN96, Lemma 4] and [Cra98], this consists of the following data:(1) The object I = ∅ .(2) For any two objects A and B , another object A ⊗ B , which we define as the concatenation of thewords A and B .(3) For any 1-morphism f : A → A (cid:48) and any object B , a 1-morphism f ⊗ B : A ⊗ B → A (cid:48) ⊗ B , whichwe define as being represented by the same word of generating morphisms as f . (This has theeffect of placing an identity tangle diagram to the right of f .)(4) For any 1-morphism д : B → B (cid:48) and any object A , a 1-morphism A ⊗ д : A ⊗ B → A ⊗ B (cid:48) , whichdefine as being represented by the same word of generating morphisms as д , except that allsubscripts are increased by the length of the word A . (This has the effect of placing an identitytangle diagram on A to the left of д .)(5) For any object B and each 2-morphism α : f → f (cid:48) , a 2-morphism α ⊗ B : f ⊗ B → f (cid:48) ⊗ B , definedas the image of α under the linear mapKhR N (cid:169)(cid:173)(cid:171) ff (cid:48) (cid:170)(cid:174)(cid:172) → KhR N (cid:169)(cid:173)(cid:173)(cid:171) ff (cid:48) B B (cid:170)(cid:174)(cid:174)(cid:172) which is induced by the link cobordism that is cylindrical, except for the a collection of disks thatcreate a collection of nested circles.(6) For any object A and each 2-morphism β : д → д (cid:48) , a 2-morphism A ⊗ β : A ⊗ д → A ⊗ д (cid:48) , definedas the image of β under the linear mapKhR N (cid:169)(cid:173)(cid:171) дд (cid:48) (cid:170)(cid:174)(cid:172) → KhR N (cid:169)(cid:173)(cid:173)(cid:171) A A дд (cid:48) (cid:170)(cid:174)(cid:174)(cid:172) which is again induced by the link cobordism that is cylindrical, except for the a collection of disksthat create a collection of nested circles.(7) For any two 1-morphisms f : A → A (cid:48) , д : B → B (cid:48) , a 2-isomorphism (cid:203) f , д : ( A ⊗ д )( f ⊗ B (cid:48) ) → ( f ⊗ B )( A (cid:48) ⊗ д ) which we define as the image of the identity 2-morphism on ( A ⊗ д )( f ⊗ B (cid:48) ) under the isotopy-induced linear map:(6.1) KhR N (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) ff дд (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) → KhR N (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) ff дд (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) It is straightforward to verify that with this data,
KhR N satisfies the axioms (i)-(viii) of a semistrictmonoidal 2-category as presented in [BN96, Lemma 4]. In fact, each axiom expresses equalities of 2-morphisms that are computed via Khovanov–Rozansky cobordisms maps, and their images are equalsince the relevant link cobordisms are isotopic. emark . The definitions of the 2-morphism spaces of
KhR N and the composition operations are moti-vated by the isomorphisms KhR N ( f , д ) (cid:27) K b ( Foam N )( (cid:110) f (cid:111) , (cid:110) д (cid:111) ) under which the horizontal compositioncorresponds to stacking tangles, the tensor product corresponds to placing tangles side by side, and thevertical composition corresponds to composing homotopy classes of chain maps. In the following, wewill take this space saving point of view when describing 2-morphisms. For example, we will say that the2-morphism in (6.1) is induced by the following movie of tangle diagrams: дf → дf A braided monoidal 2-category.
Finally, we equip
KhR N with the structure of a braided monoidal2-category. This consists of the following data:(1) The semistrict monoidal 2-category ( KhR N , ⊗ , I ) .(2) A pseudonatural equivalence R : ⊗ → ⊗ op , which assigns to pairs of objects A , B the 1-morphism R A , B : A ⊗ B → B ⊗ A given by the Morse datum of an (oriented) braid diagram of the form R A , B def = In this intentionally asymmetric braid diagram, we see boundary points A ⊗ B at the bottom and B ⊗ A at the top. Additionally, for a pair of 1-morphisms f : A → A (cid:48) , д : B → B (cid:48) , it assigns the2-isomorphism induced by the following isotopy: дf → fд (3) Additionally there is an invertible modification ˜ R −|− , − , which which associates to triples A , B , C ofobjects the 2-isomorphisms ˜ R ( A | B , C ) : ( R A , B ⊗ C )( A ⊗ R B , C ) → R A , B ⊗ C which are induced by isotopiesof the following type → Similarly, the definition of a braided monoidal 2-category calls for the existence of an invertiblemodification ˜ R − , −|− , which, however, in the case of KhR N is simply the identity modification.Using the functoriality of Khovanov–Rozansky homology it is straightforward to check that these datasatisfy the axioms of a braided monoidal 2-category as in [BN96, Definition 6].6.4. Duality.
The braided monoidal 2-category
KhR N has duals in the sense of [BMS12]. This is a slightmodification of the duality proposed by [BL03] and used by [Mac99]. Following [BMS12], instead of threedualities we only consider two dualities ∗ which correspond to rotations by π in two different axes.For an object A , the dual object A is obtained by reversing the word A and then exchanging orientations ↑↔↓ . On identity 1-morphisms, this corresponds to the result of a π rotation in a vertical line, followed bya change of orientation. There are unit and counit 1-morphisms i A : I → A ⊗ A and e A : A ⊗ A → I givenby nested collections of cups and caps, as well as a triangulator 2-isomorphism T A : ( i A ⊗ A )( A ⊗ e A ) → A represented by the obvious string-straightening isotopy. It is clear that A = A .Every 1-morphism f : A → B in KhR N has a simultaneous left and right adjoint f ∗ : B → A whichis given by the Morse data of the result of reflecting r ( f ) by π in a horizontal axis and then reversingorientations (previously we have suggestively drawn this as a reflected f in figures). Further, there are nit and counit 2-morphisms i f : A → f f ∗ and e f : f ∗ f → B , which satisfy the expected identities ( i f f )( f e f ) = f and ( f ∗ i f )( e f f ∗ ) = f ∗ . It is clear that f ∗∗ = f .For any 2-morphism α : f → д , we denote by α ∗ : д ∗ → f ∗ the 2-morphism obtained as the imageunder the isomorphism KhR N (cid:169)(cid:173)(cid:171) fд (cid:170)(cid:174)(cid:172) → KhR N (cid:169)(cid:173)(cid:171) дf (cid:170)(cid:174)(cid:172) induced by a planar anticlockwise π -rotation of the shown link diagrams. The dualities ∗ and N . The only non-trivial relation isthat for α ∈ KhR N ( f , д ) we have α ∗∗ = α , which is implicit in Definition 2.7, using the fact that foams in Foam N are considered up to isotopy relative to the boundary.6.5. Pivotality.
In [Mac99] Mackaay introduces the notion of sphericality for monoidal 2-categorieswith suitable duals. This boils down to the extra structure providing natural 2-isomorphisms betweenright- and left-traces of 1-endomorphisms. f → f For a braided monoidal 2-category with duals, such as
KhR N , which is categorified ribbon in the sense thatit admits 2-isomorphisms that provide a vertical categorification of the framed Reidemeister I move , suchisomorphisms always exist. In fact there are two natural choices, corresponding to sliding the closurearcs over or under the diagram for f : f → f → f → f , f → f → f → f The sweep-around property implies that these two choices produce equal 2-isomorphisms in
KhR N .(Compare [HPT16, Prop A.4], for an apparently analogous situation one dimension down.)Motivated by the equivalent fact that KhR N carries a well-defined action of Diff + ( S ) , we propose that KhR N should be a prototypical example of some future definition of a SO ( ) -pivotal braided monoidal2-category, and suggest the possibility that these are the SO ( ) fixed points in the braided monoidal2-categories with duals. Remark . An analogous trigraded semistrict braided monoidal dg 2-category
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