gl(2) foams and the Khovanov homotopy type
ggl FOAMS AND THE KHOVANOV HOMOTOPY TYPE
VYACHESLAV KRUSHKAL AND PAUL WEDRICHA
BSTRACT . The Blanchet link homology theory is an oriented model of Khovanovhomology, functorial over the integers with respect to link cobordisms. We formulatea stable homotopy refinement of the Blanchet theory, based on a comparison of theBlanchet and Khovanov chain complexes associated to link diagrams. The construc-tion of the stable homotopy type relies on the signed Burnside category approach ofSarkar-Scaduto-Stoffregen.
1. I
NTRODUCTION
The Khovanov homology [Kho00] is an invariant of oriented links in R . Givensuch a link L , the associated invariant Kh( L ) is a bigraded abelian group. A linkcobordism Σ in R × [0 , induces a map on Khovanov homology which is well-defined up to a sign, see Jacobsson [Jac04]. The map is defined by decomposing Σ into elementary cobordisms, corresponding to Morse surgeries and Reidemeistermoves on link diagrams; decompositions of isotopic link cobordisms are related bythe movie moves of Carter–Saito [CS93], and some of them indeed change the sign ofthe induced map [Jac04, Lemma 5.2].Several models have been formulated (Blanchet [Bla10], Caprau [Cap08], Clark-Morrison-Walker [CMW09], Sano [San], Vogel [Vog20]) to fix the sign ambiguityfor cobordism maps in Khovanov homology, while producing isomorphic link ho-mology groups. The resulting full functoriality is important in particular for the -dimensional aspect of the Khovanov theory [MWW19] and topological applications.An entirely different refinement of Khovanov homology is due to Lipshitz andSarkar. In [LS14a] they associate a suspension spectrum X ( L ) to each link diagram L ,whose stable homotopy type is an invariant of links in R and whose cohomology isthe Khovanov homology of L . It is shown in [LS14b] that a link cobordism Σ : L −→ L in R × [0 , represented as a sequence of Reidemeister moves and elementaryMorse cobordisms gives rise to a map of spectra X ( L ) −→ X ( L ) , whose inducedmap on cohomology is the Khovanov map Kh(Σ) : Kh( L ) −→ Kh( L ) . The map ofspectra associated to link cobordisms is conjectured [LS14b] to be well-defined, upto an overall sign – like the map on Khovanov homology.In this paper we combine the two theories by constructing a stable homotopy re-finement X or ( L ) of the Blanchet theory [Bla10]. Theorem 1.1.
Let L be an oriented link diagram. The stable homotopy type X or ( L ) is aninvariant of the isotopy class of the link. Its cohomology is isomorphic to Blanchet’s orientedmodel of Khovanov homology of L . a r X i v : . [ m a t h . G T ] J a n VYACHESLAV KRUSHKAL AND PAUL WEDRICH
Our first motivation to pursue this result is the goal of developing a fully functorialstable homotopy invariant for links in R and their cobordisms. To this end, ourconstruction fixes the sign ambiguity, like the Blanchet theory does for Khovanovhomology. Building on [LS14b], we show (Lemma 4.5) that a link cobordism L −→ L , presented as a sequence of elementary cobordisms, gives rise to a map of spectra X or ( L ) −→ X or ( L ) . The induced map on cohomology is the map given by theBlanchet theory. Our methods do not address the conjecture that the map inducedby a link cobordism Σ is well-defined with respect to isotopies of Σ .A strategy for proving Theorem 1.1 should be well-known to the (hitherto possiblyempty) intersection of two communities of experts. It relies on two technical tools: • a choice of a natural (though not necessarily canonical) isomorphism betweenthe Khovanov and Blanchet theories, e.g. as elaborated by Ehrig–Stroppel–Tubbenhauer and Beliakova–Hogancamp–Putyra–Wehrli [EST16; Bel+19], and • the signed Burnside category framework of Sarkar–Scaduto–Stoffregen from[SSS20], which accommodates such natural isomorphism,to lift the reformulation of the Khovanov homotopy in terms of the Burnside cate-gory from Lawson–Lipshitz–Sarkar [LLS20; LLS17] to Blanchet’s setting. The expo-sition in this paper follows the strategy outlined above, starting from the perspectiveof the Blanchet theory. In particular, we do not assume the reader to be familiar withthe technical tools mentioned above.Our emphasis on giving explicit constructions using the combinatorics of gl websand foams is related to our second motivation, namely the problem of extending theLipshitz–Sarkar construction to sl N (or, more accurately, gl N ) homology theories for N ≥ . The framed flow category construction of the Khovanov homotopy typein [LS14a], as well as its reformulation in terms of the Burnside category [LLS20;LLS17] utilize the existence of canonical generators of the Khovanov chain complexassociated to a link diagram. Moreover, the components of the Khovanov differentialhave coefficients , with respect to these generators. As such, the differential maybe combinatorially encoded using correspondences in the context of the Burnsidecategory.The gl N link homology theories are more intricate; in particular both the prob-lem of identifying a basis and computing the differential are substantially more in-volved. The Blanchet theory, which fits in the foam category formulation of gl N link homology theories at N = 2 , is thus an interesting test case. Given an orientedlink diagram, we use an additional combinatorial piece of data associated to the linkdiagram, a flow with values in { +1 , − } , to construct a canonical set of gl foam gen-erators (see Section 3). The differential has coefficients in the set { , ± } with respectto these generators. Importantly, Proposition 3.13 shows that the edge differentialand the cobordism maps are sign-coherent , in the sense that for any two webs V, W related by a zip/unzip or saddle foam, the associated map has coefficients in either { , } or { , − } with respect to the chosen bases. These results are then used in See [RW20; ETW18], building on [MSV09; LQR15; QR16]. l FOAMS AND THE KHOVANOV HOMOTOPY TYPE 3
Section 4 to formulate the stable homotopy type X or ( L ) and to define the maps ofspectra induced by elementary link cobordisms. The construction is given using thesigned Burnside category framework of [SSS20]. In Section 4.5 we show that X or ( L ) is homotopy equivalent to X ( L ) of [LS14a].Our construction of X or ( L ) can be seen as a step towards a construction of stablehomotopy refinements of the gl N link homology theories using framed flow categorymethods. However, there are several features that make the analysis substantiallyeasier in the N = 2 theory considered here, and new ideas are needed to addressthe N > case. See the work of Jones–Lobb-Sch ¨utz [JLS19] in this direction. Itwould also be interesting to compare the present work with Kitchloo’s constructionsin [Kit19].Constructing a gl version of the tangle invariant of [LLS] is outside the scope ofthis paper, but we expect that the methods considered here should be helpful forformulating it as well. We also hope that our results will contribute to the develop-ment of a stable homotopy refinement of the 4-manifold invariants from [MWW19];indeed this was a main motivation for our work. Acknowledgements.
We would like to thank Robert Lipshitz and Sucharit Sarkarfor helpful discussions related to this work. The first named author also would liketo thank Rostislav Akhmechet and Michael Willis for many conversations on thesubject of stable homotopy refinement of link homology theories.VK was supported in part by the Miller Institute for Basic Research in Scienceat UC Berkeley and Simons Foundation fellowship 608604. PW was supported bythe National Science Foundation under Grant No. DMS-1440140, while in residenceat the Mathematical Sciences Research Institute in Berkeley, California, during theSpring 2020 semester.2. B
LANCHET ’ S ORIENTED MODEL FOR K HOVANOV HOMOLOGY
Here we recall the construction of Blanchet’s oriented model of Khovanov homol-ogy using gl foams. We shall assume that the reader is familiar with Bar-Natan’sdescription of Khovanov homology using dotted cobordisms from [Bar05]. Definition 2.1.
The dotted cobordism category
Cob is the additive completion of thegraded pre-additive category with • objects given by closed unoriented 1-dimensional manifolds properly embed-ded in R (and grading shifts thereof, encoded by powers of q ), • morphisms given by Z -linear combinations of dotted cobordisms, i.e. unori-ented 2-dimensional manifolds properly embedded in R × I , whose compo-nents may be decorated with dots , considered up to isotopy rel boundary andthe local relations(1) = 0 , • = 1 , = • + • , •• = 0 . VYACHESLAV KRUSHKAL AND PAUL WEDRICH • composition is given by (the bilinear extension of) stacking cobordisms, • the grading requires a dotted cobordism C : q k M −→ q l M to satisfy dots − χ ( C ) = l − k. This cobordism category categorifies the Temperley–Lieb skein theory of linearcombinations of isotopy classes of unoriented planar curves, modulo the relation (cid:13) = ( q + q − ) ∅ , which models the pivotal tensor category of U q ( sl ) -modules generated by the natu-ral 2-dimensional module.Next we describe the category of Blanchet foams as an oriented extension of Cob .Blanchet foams categorify the skein theory of gl webs, which models the pivotaltensor category of U q ( gl ) -modules generated by the natural 2-dimensional moduleand its exterior square, see [CKM14; QS19; TVW17].Here we define a gl web to be an oriented, trivalent graph properly embedded in R , with a flow on the edges taking values in the set { , } . Practically, this meansa web is a graph drawn in the plane, with oriented edges labelled either or (anddrawn solid or doubled respectively) and at vertices two -labelled edges mergeinto a -labelled one, or a -labelled edge splits into two -labelled ones. (Examplesof (local) webs are shown in Lemma 3.1 and Lemma 3.2.)We also need the notion of (embedded) gl foams, which form the morphisms inBlanchet’s category. Foams for gl are “trivalent surfaces” which we describe next,properly embedded in R × I , equipped with a flow taking values in { , } . Foamsare assembled from compact oriented - or -labelled surfaces, called facets , whichare glued along (part of) their boundary so that precisely two boundary componentsof -labeled facets are identified with a single boundary component of a -labeledfacet with the opposite orientation. Points on such seams have a neighbourhoodmodelled on the letter Y times an interval, and all other points have disk neighbour-hoods . Facets with label are furthermore allowed to carry dots.Given a gl foam F or web W , we denote by c ( F ) the unoriented surface obtainedby deleting all 2-labelled facets in F , and similarly c ( W ) denotes the result of deletingall -labelled edges in W . Definition 2.2.
The gl foam category Foam is the additive completion of the gradedpre-additive category with: • objects given by webs embedded in R (and grading shifts thereof), • morphisms given by Z -linear combinations of gl foams properly embeddedin R × [0 , , considered up to isotopy rel boundary and Blanchet’s local re-lations [Bla10], which include (1) on 1-labelled facets as well as: Or, rather, its extension to a certain type of 3-category termed “canopolis” by Bar-Natan Foams for gl N for N ≥ require other singularities, which can be avoided here. l FOAMS AND THE KHOVANOV HOMOTOPY TYPE 5 (2) = − , = − , •• αβ = δ α, δ β, − δ α, δ β, (3) = , = • − • Here we shade -labelled facets blue and -labelled facets yellow . An es-sential feature of Blanchet’s relations is that swapping all orientations intro-duces a minus sign in both of the relations in (3). • composition given by (the bilinear extension of) stacking foams, • the grading requires a foam F : q k W −→ q l W to satisfy dots − χ ( c ( F )) = k − l. In particular, for any objects W , W , the morphism spaces Hom
Foam ( W , W ) are abelian groups. To capture morphisms of various degrees, we will alsoconsider the graded abelian groups: HOM
Foam ( W , W ) := (cid:77) k Hom
Foam ( q − k W , W ) We record one important consequence of these relations, namely the dot-slidingrelation(4) • = − • . The presentation of the relation (1) and (3) is imported from Lauda–Queffelec–Rose [LQR15], where it is shown that Blanchet’s foam relations also arise in certainquotients of categorified quantum groups of type A. For a generalisation to foamsfor gl N , see Robert–Wagner [RW20]. Remark 2.3.
The last relation in (1) (and thus also (4)) can be deformed to yielda foam-based construction of Lee’s deformed Khovanov homology [Lee05] or anequivariant link homology [Kho06]. For a careful study of such deformations, see[EST16; KK20]. For the gl N case see [RW16]. Remark 2.4.
After specialising the ground ring from Z to F , the map which deletes -labelled facets and edges induces a full, essentially surjective functor c : Foam F −→ Cob F , In grayscale, blue appears darker than yellow.
VYACHESLAV KRUSHKAL AND PAUL WEDRICH as can be seen by comparing Blanchet’s foam relations with Bar-Natan’s cobordismrelations. In fact, c is also faithful. For refined versions of this statement see Queffelec–Wedrich [QW18b] and Beliakova–Hogancamp–Putyra–Wehrli [Bel+19].2.1. Blanchet’s oriented model.
To crossings in oriented link diagrams, Blanchet[Bla10] associates chain complexes of (local) webs and foams: (cid:114) (cid:122) or := (cid:32) −→ q − (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) −→ q − → (cid:33) , (cid:114) (cid:122) or := (cid:32) → q → q (cid:58)(cid:58)(cid:58)(cid:58)(cid:58) → (cid:33) with the (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) underlined terms in homological degree zero and with differentials mod-elled by the so-called zip and unzip foams, cf. [Bla10, Section 3.2]: , For a link diagram L (with an ordering of the crossings), the chain complex (cid:74) L (cid:75) or is defined as the total complex of the hyper cube of resolutions , obtained by resolvingeach crossing locally as shown above.As in Bar-Natan’s description of Khovanov homology, when considered as anobject of the bounded homotopy category of chain complexes of webs and foams,denoted K b ( Foam ) , the complex (cid:74) L (cid:75) or is an invariant of the link represented by L ,well-defined up to isomorphism. In particular, Reidemeister moves induce chainhomotopy equivalences between the associated complexes.The passage from (cid:74) L (cid:75) or to a chain complex of graded abelian groups proceeds inanalogy with Khovanov’s original theory, using a TQFT for webs and foams that cannow be described as a representable functor.The Blanchet–Khovanov chain complex of L is the chain complex of graded abeliangroups defined as(5) CKh or ( L ) := HOM • Foam ( ∅ , (cid:74) L (cid:75) or ) where HOM • Foam ( ∅ , (cid:74) L (cid:75) or ) denotes the chain complex of bihomogeneous maps be-tween the Z × Z -graded objects ∅ and (cid:74) L (cid:75) or , with the differential induced by thedifferential on the target (the source has trivial differential).The oriented (or gl ) Khovanov homology of L is the bigraded abelian group Kh or ( L ) := H • (CKh or ( L )) . It follows from the discussion above that Kh or ( L ) is an invariant of the link rep-resented by L , well-defined up to isomorphism of bigraded abelian groups. In fact,these isomorphisms can be chosen coherently, as shown by Blanchet’s main result,which we paraphrase as follows. Theorem 2.5.
Let C be an oriented smooth link cobordism, properly embedded in R × I ,between oriented links represented by diagrams L and L (cid:48) . Suppose that C is in genericposition, so that it can be represented by a finite sequence of Reidemeister and Morse moves, l FOAMS AND THE KHOVANOV HOMOTOPY TYPE 7 transforming L into L (cid:48) . There exists an assignment of Reidemeister isomorphisms and Morsemaps on the level of Kh or , so that the composite map Kh or ( C ) is invariant under isotopy relboundary on C . As a consequence, gl Khovanov homology constitutes a functor: Kh or : (cid:26) links embedded in R link cobordisms in R × I /isotopy (cid:27) −→ gr Z × Z Abgrp
Other functorial versions of Khovanov homology have been constructed by Caprau [Cap08],Clark–Morrison–Walker [CMW09], Sano [San], and Vogel [Vog20]. Blanchet’s ap-proach to functorial link homologies via foams works over the integers and is dis-tinguished in that it extends to gl N , see Ehrig–Tubbenhauer–Wedrich [ETW18], andto links in S , see Morrison–Walker–Wedrich [MWW19].3. B ASES FOR WEB SPACES
The goal of this section is to introduce generators of the webs spaces, and to ana-lyze the coefficients of the Blanchet-Khovanov differential, as well as of the elemen-tary link cobordisms with respect to these generators. These results underlie theformulation of the stable homotopy type in Section 4.3.1.
Construction of bases.
The following local relations hold in the category
Foam and will be used throughout this section to simplify webs and foams.
Lemma 3.1.
There are isomorphisms between webs in
Foam which differ only in a disk asshown (or as shown, but with all orientations reversed): ∼ = q ∅ ⊕ q − ∅ , ∼ = ∅ (6) = , ∼ = (7) Proof.
As in [Bla10]. (cid:3)
Lemma 3.2.
For every n ≥ , there is an isomorphism in Foam that simplifies the coherentlyoriented (clockwise or anti-clockwise) n -gon web W n , e.g.: n =1 ∼ = , n =2 ∼ = , n =3 ∼ = The isomorphism can be chosen to be a foam F n , such that c ( F n ) = c ( W n ) × [0 , and suchthat F n contains a single -labeled disk in the region shown.Proof. Use the local isomorphisms from (7) to undock the 2-labeled edges from theboundary edges, leaving a central 2-labeled circle, which can then be removed via(6). (cid:3)
The isomorphisms in Lemma 3.2 simplify webs by removing coherently (anti-)clockwise oriented regions by undocking 2-labeled edges on the left (right) side ofthe remaining 1-labeled edges. The following lemma shows that this is possible forevery closed web, if we also allow exceptional cases of the following type:
VYACHESLAV KRUSHKAL AND PAUL WEDRICH W −→ W Here we consider the outside region as a coherently clockwise oriented and per-form undocking towards the left. The resulting 2-labeled outside circle can be re-moved by a cap foam after simplifying the remaining web nested inside it, as de-scribed in the following lemma.
Lemma 3.3.
Every closed gl web W is isomorphic to a direct sum of grading shifts ofthe empty web via an isomorphism which is composed of exclusively clockwise local isomor-phisms from Lemma 3.2 (including ‘outside’ versions) and the circle removal isomorphisms (6) . As such, this isomorphism is unique. The statement with exclusively anti-clockwiseisomorphisms holds analogously.Proof. The statement is trivial for webs W without trivalent vertices, so let us assumethat W has trivalent vertices. Moreover, we will assume that W is connected, forotherwise we can treat the connected components independently, starting with aninnermost one with respect to the nesting condition. Differences in ordering thesesimplifications give the same result due to far-commutativity.Fix a base point p ∈ R \ W in the outside region . We now start by labeling eachregion in R \ W with its ‘winding number’ around p . For this, pick a point q in theregion and a path γ : [0 , −→ R with γ (0) = q and γ (1) = p . Then we label theregion containing q by the algebraic intersection number of γ with W . (Here eachsigned intersection with the -labeled edges counts as ± .) E.g. the region enclosedby a clockwise circle will have winding number . The outside region has windingnumber , by definition.Now we have two cases to consider. If the maximal winding number of regionsis positive, then each maximal region has a coherently clockwise oriented boundaryand can be removed via Lemma 3.2.Otherwise, the maximal winding number is zero, and the outside region attainsthis maximum. This implies that the outermost cycle in the web (the boundary ofthe region containing p ) is coherently counter-clockwise oriented. In this case, weuse the ‘outside version’ of the undocking move to simplify the web.This algorithm terminates since it strictly reduces the number of trivalent verticesin every step. Since regions of maximal winding number cannot be adjacent, theorder of simplifying is again irrelevant due to far-commutativity. (cid:3) Lemma 3.3 allows us to build a basis for the vector space
HOM
Foam ( ∅ , W ) for anyweb W . One of the possible variations still allowed are rescalings of the circle re-moval isomorphisms from (6) by signs. To simplify local computations we choose arescaling depending of the auxiliary data of a flow, that we now introduce. l FOAMS AND THE KHOVANOV HOMOTOPY TYPE 9
Definition 3.4. A flow f with values { +1 , − } on a web W in Foam assigns to every1-labeled edge of W an element of { +1 , − } and to every 2-labeled edge of W theentire set { +1 , − } , such that every trivalent vertex in W has adjacent labels as in thefollowing: +1 − { +1 , − } , − { +1 , − } , +1 − { +1 , − } , − { +1 , − } I.e. the flow condition is satisfied at every vertex. Similarly, a flow on a foam asso-ciates to 1-labeled facets elements of { +1 , − } and to 2-labeled facets the entire set { +1 , − } , such that the flow condition is satisfied at every seam. A flow on a foaminduces a flow on its boundary webs. When considering a foam F : W −→ V be-tween webs with flows f W and f V , then we say these flows are compatible with F ifthere exists a flow f on F that restricts to f W and f V on the corresponding boundarycomponents.We say a flow on a web W is admissible if every non-closed 2-labeled edge has aneighborhood of the following two types:(8) +1 − − , − − In words, the flow has to stay parallel, and not cross, at 2-labeled edges.
Lemma 3.5.
Let L be an oriented link diagram in R , which we consider as a union ofarcs that connect crossings and disjoint closed components. Next we color the connectedcomponents of R \ L with the checkerboard black and white coloring, starting with white onthe unique unbounded component. On every oriented arc and every closed component of L ,we now place the label +1 if there is a white component on the left, and the label − otherwise.This labelling descends to an admissible flow on every web W in the cube of resolutions of L ,compatible with the foams representing the components of the differential.Moreover, this assignment of a canonical flow to each link diagram in R is compatiblewith split disjoint union of link diagrams.Proof. At the site of every crossing, the webs W contain either a parallel resolutionor a 2-labelled edge. In the latter case, the labels are in the configurations of (8), de-pending on whether there is a white or black region to the left of this crossing in L . Inthe case of a parallel resolution, the situation is analogous. Thus, the labels assembleto flows which are admissible at 2-labelled edges and compatible with the zip andunzip foams realising the components of the differential in the cube of resolutions.The statement about compatibility of the flow construction with split disjoint unionfollows from the fact that this operation embeds two white outside regions into awhite outside region. (cid:3) Remark 3.6.
More generally, using the same arguments as in Lemma 3.5 one canshow that any foam between closed gl webs in R × [0 , admits a canonical flow. Definition 3.7.
Let W be a gl web with an admissible flow f . We define a basis B ( W, f ) of HOM
Foam ( ∅ , W ) containing (signed) foams appearing as entries of theinverse of the isomorphism obtained in Lemma 3.3. These foams are determined bythe local simplifications in Lemma 3.2, where we only use clockwise regions, andthe isomorphisms realising (6). To determine the signs, note that the flow f on W extends uniquely to a flow on every basis foam, which we again denote by f . Wenow equip the basis foams with a minus sign if they contain an odd number of dotson facets where f has value − . Example 3.8.
The bases for 1-labeled circles with flow +1 and − are: B ( (cid:13) , +1) = (cid:110) +1 , • +1 (cid:111) , B ( (cid:13) , −
1) = (cid:110) − , − • − (cid:111) We will write and X for the undotted and dotted cup foam, respectively. Bothbases above can then be written as { , (cid:15) X } . Example 3.9.
For the theta web, there again exist two flows, whose associated basesare: B (cid:16) +1 − (cid:17) = (cid:110) +1 , • +1 (cid:111) , B (cid:16) − (cid:17) = (cid:110) − , − • − (cid:111) Let us call the theta web with the first flow the left theta web and the other one the right theta web . Example 3.10.
The unzip foam from the left theta web to two circles has all its coeffi-cients in the set { , } in the above bases. This is seen using the neck-cutting relation(1) and the last relation in equation (2). For the right theta web, it has coefficients { , − } .The zip foam with target the theta web (with any flow) and source given by twocircles with compatible flows has coefficients { , } in the above bases.3.2. Action of foams on bases.
Here we study the action of elementary foams onthe web bases B ( W, f ) constructed in the previous section. To this end, we will needtwo auxiliary results. Proposition 3.11. (Preparing generalized neck-cutting) Let F be a foam in R × [0 , whoseunderlying 1-labeled surface c ( F ) has a compression disk D . Then there exist tubular neigh-bourhoods D ⊂ U ⊂ V ⊂ R × (0 , and a foam G in R × [0 , such that • G ∩ ( R \ V ) = F ∩ ( R \ V ) • c ( G ) = c ( F ) • c ( F ∩ V ) ∼ = ∂D × [0 , • G ∩ U ∼ = ∂D × [0 , • F = ± G as morphisms in Foam l FOAMS AND THE KHOVANOV HOMOTOPY TYPE 11
To paraphrase this: any foam, whose underlying 1-labeled surface has a neck, canbe modified locally and up to sign into a new foam, which itself has a 1-labeled neck.
Proof.
This was shown in the proof of [QW18b, Lemma 3.6], so we only give a sketchhere. Assuming that D intersects F generically, we can find a slightly larger opendisk D (cid:48) and a tubular neighbourhood V ∼ = D (cid:48) × (0 , , such that F ∩ V ∼ = W × (0 , forsome gl web W with c ( W ) = ∂D . Using the 2-labeled circle, saddle, and undockingrelations from 3.1, one can see that the foam W × (0 , equals up to a sign a foam G (cid:48) that factors through a web W (cid:48) with ∂W (cid:48) = ∂W , c ( W (cid:48) ) = c ( W ) and D ∩ W (cid:48) = ∂D .Detailed descriptions of this process appear in [QW18b, Lemmas 2.1 and 3.6] and[QW18a, Lemma 64]. The foam G is now obtained by gluing G (cid:48) and F ∩ ( R \ V ) . (cid:3) Proposition 3.12. If F and G are (possibly dotted) Blanchet foams with identical underlying(dotted) surfaces c ( F ) = c ( G ) and ∂F = ∂G , then F = ± G .Proof. This was proven in [Bel+19, Proposition 2.9]. Here we sketch an alternativeproof based on Proposition 3.11. First we may assume that W = ∂F = ∂G ⊂ R ×{ } ,for otherwise we would bend the bottom boundary of F and G to the top.Fix an admissible flow f on W and consider the basis B ( W, f ) ⊂ HOM
Foam ( ∅ , W ) .By construction, all elements of B ( W, f ) are given by decorating the 1-labeled com-ponents of a distinguished foam F W by certain signed dots. We will also considera basis B ( W, f ) ∗ of HOM
Foam ( W, ∅ ) which is dual to B ( W, f ) under the compositionpairing valued in END
Foam ( ∅ ) = Z (cid:104)∅(cid:105) ∼ = Z . More specifically, we take B ( W, f ) ∗ to consist of foams obtained by appropriately decorating a distinguished foam F W with signed dots ( F W can be obtained by reflecting F W in R × { } .Now we claim, there exists (cid:15) ∈ {± } such that H ◦ F = (cid:15) ( H ◦ G ) for any H ∈ B ( W, f ) ∗ . This would imply F = (cid:15)G .To prove the claim, we choose base points on 1-labeled facets of F W , such thatevery non-closed connected component of c ( F ) = c ( G ) contains exactly one suchbase point up inclusion into c ( F W ◦ F ) = c ( F W ◦ G ) . The foams H ◦ F and H ◦ G aresigned, dotted versions of F W ◦ F and F W ◦ G respectively, and to verify their equalitymodulo foams relations up to a global sign, we may assume that all dots comingfrom H and from non-closed components of c ( F ) = c ( G ) have been transported tothe base points. This is possible, since the dots originating in F and G are moved tothe base points at the expense of a global sign (cid:15) (cid:48) , irrespective of H .We have now reduced the problem to a comparison of foams C ◦ d H ◦ F (cid:48) and (cid:15) (cid:48) C ◦ d H ◦ G (cid:48) where C consists entirely of 1-labeled caps, d H is a signed, dotted identity foam on ∂C , with signs and dots depending on H ∈ B ( W, f ) ∗ , F (cid:48) and G (cid:48) are foams satisfying c ( F (cid:48) ) = c ( G (cid:48) ) and containing dots only on closed components of c ( F (cid:48) ) = c ( G (cid:48) ) . Inparticular, we have eliminated the sign dependence on H and are left with proving F (cid:48) = ± G (cid:48) .Since c ( F (cid:48) ) = c ( G (cid:48) ) we have the same necks in F (cid:48) and G (cid:48) , which can be cut us-ing Proposition 3.11 and the neck-cutting relation from (1) at the expense of a global sign. Similarly, every S component in the underlying surfaces of the result can be re-moved by another application of 3.11 and the sphere relations from (1). See [QW18b,Lemma 3.9] for a detailed description of this step. After these simplifications, wemay assume that F (cid:48) = (cid:15) F (cid:80) i d i F (cid:48)(cid:48) and G (cid:48) = (cid:15) G (cid:80) i d i G (cid:48)(cid:48) where F (cid:48)(cid:48) and G (cid:48)(cid:48) are un-dotted foams such that c ( F (cid:48)(cid:48) ) = c ( G (cid:48)(cid:48) ) consists entirely of cups (disks), d i is a dottedidentity foam on ∂C and (cid:15) F , (cid:15) G ∈ {± } . Here we have used in a crucial way that theneck-cutting relation from (1) is sign-coherent and always involves foam facets withthe same flow label. Finally, we observe that F (cid:48)(cid:48) = ± G (cid:48)(cid:48) by undocking and elimi-nating all 2-labeled facets in both foams via the first two relations in (2) and the firstrelation in (3). (cid:3) Next, we compute the action of zips, unzips, and saddle maps between standardbases. Note that a saddle cobordism between link diagrams L and L (cid:48) induces saddlefoams between webs W and W (cid:48) appearing in the respective cubes of resolutions,which are compatible with the canonical flows defined in Lemma 3.5. Proposition 3.13.
Let W be a web with an admissible flow f , and let F : W −→ V denotea zip or an unzip foam or a saddle to a web V , which we equip with the induced flow, denoted g . Then the matrix of the linear map F ∗ : HOM Foam ( ∅ , W ) −→ HOM
Foam ( ∅ , V ) , F ∗ ( G ) = F ◦ G with respect to the bases B ( W, f ) and B ( V, g ) has entries in either { , } or { , − } . This says that the action of F ∗ is sign-coherent with respect to the standard bases. Proof.
Recall that every basis element H ∈ B ( W, f ) is a signed, dotted version of adistinguished foam F W ∈ HOM
Foam ( ∅ , W ) such that c ( F W ) is a collection of disks.Let us write H = d H ◦ F W , where d H is a signed, dotted identity foam on W , withsigns determined by the flow f . Then we have: F ◦ H = F ◦ d H ◦ F W = d (cid:48) H ◦ F ◦ F W for a signed, dotted identity foam on V with signs determined by the flow g .Suppose that c ( F ) is a saddle that merges two circles in c ( W ) (recall that F itselfmay be a zip, an unzip, or a saddle). Then we have c ( F ◦ F W ) = c ( F V ) and ∂ ( F ◦ F W ) = ∂F V , and thus by Proposition 3.12, F ◦ F W = (cid:15) · F V for some (cid:15) ∈ {± } .Furthermore, if d (cid:48) H ◦ F ◦ F W = (cid:15) · d (cid:48) H ◦ F V is non-zero, then it agrees with an elementof B ( V, g ) up to the global sign (cid:15) .Now suppose that c ( F ) is a saddle that splits two circles in c ( W ) . In this case, wecan find a compression disk D for c ( F ◦ F W ) to which we apply Proposition 3.11 andthe neck-cutting relation from (1). The result will be a linear combination (cid:15) (cid:48) ( d ◦ G V + d ◦ G V ) where (cid:15) (cid:48) ∈ {± } , G V is an undotted foam with c ( G V ) = c ( F V ) , and d and d are both identity foams on V with a single dot placed on certain facets with thesame flow label (cid:15) (cid:48)(cid:48) ∈ {± } . By Proposition 3.12 we conclude that G V = (cid:15) (cid:48)(cid:48)(cid:48) F V forsome (cid:15) (cid:48)(cid:48)(cid:48) ∈ {± } , and thus F ◦ F W = (cid:15) (cid:48) (cid:15) (cid:48)(cid:48)(cid:48) ( d ◦ F V + d ◦ F V ) . Finally, for other basiselements we compute d (cid:48) H ◦ F ◦ F W = (cid:15) (cid:48) (cid:15) (cid:48)(cid:48)(cid:48) ( d (cid:48) H ◦ d ◦ F V + d (cid:48) H ◦ d ◦ F V ) . Whenever l FOAMS AND THE KHOVANOV HOMOTOPY TYPE 13 these summands are nonzero, they agree with a basis element from B ( V, g ) up to theglobal sign (cid:15) (cid:48) (cid:15) (cid:48)(cid:48) (cid:15) (cid:48)(cid:48)(cid:48) . (cid:3) Finally, we consider the effect of the cup and cap cobordisms.
Lemma 3.14.
Let W be a web with an admissible flow f , containing an innermost 1-labelledcircle C with flow label (cid:15) . Let F denote the cap foam that removes this component, resultingin a web V := W \ C that we equip with the induced flow g . Further, let F ! : V −→ W denote the corresponding cup foam in the opposite direction. Then the matrix of the linearmap F ∗ : HOM Foam ( ∅ , W ) −→ HOM
Foam ( ∅ , V ) , F ∗ ( G ) = F ◦ G with respect to the bases B ( W, f ) and B ( V, g ) has entries in { , (cid:15) } and the matrix of thelinear map F ! ∗ : HOM Foam ( ∅ , V ) −→ HOM
Foam ( ∅ , W ) , F ! ∗ ( G ) = F ! ◦ G with respect to the bases B ( V, g ) and B ( W, f ) has entries in { , } .Proof. Immediate from Example 3.8 and the relations (1). (cid:3)
Remark 3.15.
In this section we chose the additional local data of flows on websand foams to record and track the sign discrepancies between Bar-Natan cobordismsand Blanchet foams, which are essential for the functoriality of Khovanov homologyand link cobordisms. The use of flows is motivated by the Murakami–Ohtsuki–Yamada state-sum for evaluation of closed webs [MOY98] and its analog for foamsthat is due to Robert–Wagner [RW20]. Other, essentially equivalent ways of encod-ing these sign discrepancies include skein-algebra like superposition operations with2-labelled curves [QW18b] and shadings [Bel+19].
Remark 3.16.
The results of this section also hold for the deformed foam categoriesmentioned in Remark 2.3. When deforming the relation • = 0 in (1) to • − h • + t = 0 ,the signed dots −• in basis elements should be replaced by the linear combination ◦ := h − • , see [Bel+19]. 4. T HE STABLE HOMOTOPY TYPE
The signed Burnside category and the construction of X or ( L ) . A stable homo-topy refinement of Khovanov homology was originally introduced by Lipshitz andSarkar in [LS14a]. Their construction was reformulated by Lawson, Lipshitz andSarkar [LLS20] using the Burnside category . In this section we formulate the stablehomotopy type X or ( L ) in Theorem 1.1. Our construction relies on the signed Burnsidecategory , introduced by Sarkar, Scaduto and Stoffregen in [SSS20] in their work onthe odd Khovanov homotopy type.Recall from Section 2.1 the setup of the Blanchet theory: given an oriented link dia-gram L with n crossings, there is a cube of resolutions (cid:74) L (cid:75) or and the chain complex ofgraded abelian groups CKh or ( L ) . More precisely, to each vertex v of the cube { , } n See also [HKK16] there is an associated planar web W ( v ) , and each edge corresponds to a zip/unzipfoam. Lemma 3.5 gives a canonical flow f can on every web W ( v ) in the cube ofresolutions of L , compatible with the foams representing the components of the dif-ferential. A basis B ( v ) for the web space at each vertex v is constructed in Definition3.7. Finally, Proposition 3.13 determines the coefficients of the edge differential withrespect to the chosen bases.Several variations of the stable homotopy type construction using different ver-sions of the Burnside category have appeared in the literature; therefore we willoutline steps of the construction rather than giving a detailed exposition. We willrefer to statements in [LLS20; SSS20] and emphasize details of our setting that aredifferent from these references.Consider the cube category n whose objects are elements of { , } n and with aunique morphism φ a,b iff a ≥ b (here the partial ordering on the objects is induced bythe ordering of the coordinates). The construction is based on the lift F or (9) B σ n Z − Mod F opor F or to the signed Burnside category B σ (the definition is recalled below) of the cube ofresolutions, viewed as a diagram of abelian groups, F opor : 2 n −→ Z − Mod . Here thesubscript stands for “oriented”, and the superscipt reflects the fact that the arrowsin n are dual to those in the usual cube of resolutions. The value of F opor on verticesand edges of the cube is defined by the representable functor HOM
Foam ( ∅ , − ) as inequation (5). The construction of the Khovanov stable homotopy type in [LLS20]relied on the Burnside category B , a weak 2-category whose objects are finite sets, -morphisms are finite correspondences, and -morphisms are maps of correspon-dences. We use the signed Burnside category B σ , introduced in [SSS20], where -morphisms are signed correspondences discussed in more detail below. This exten-sion is needed to accommodate the signs appearing in the differential in the orientedmodel for Khovanov homology (see Proposition 3.13). The vertical map in the dia-gram (9) is the forgetful functor sending a finite set to the abelian group generatedby it.There is a forgetful functor B σ −→ B , and the functor F or constructed in this paperis a lift of F Kh : 2 n −→ B of [LLS20]. (A different lift, F o in [SSS20], corresponds tothe odd Khovanov homology.) As in prior constructions, the functor F or decomposesas (cid:96) j F j or along quantum gradings.Given finite sets X, Y , a signed correspondence [SSS20, Section 3.2] is a tuple ( A, s A , t A , σ A ) where A is a finite set, the source and target maps s A : A −→ X, t A : A −→ Y aremaps of sets, and σ A : A −→ {± } is a “sign map”. Signed correspondences are l FOAMS AND THE KHOVANOV HOMOTOPY TYPE 15 conveniently encoded as diagrams(10) {± } AX Y s A σ A t A The following lemma is a useful tool for constructing functors to the signed Burn-side category.
Lemma 4.1. ([LLS20, Lemma 4.4], [SSS20, Lemma 3.2])
The following data satisfyingconditions (1), (2) can be extended to a strictly unitary lax -functor F : 2 n −→ B σ , whichis unique up to natural isomorphism:A finite set F ( u ) for each vertex u ∈ n , a signed correspondence F ( ϕ u,v ) from F ( u ) to F ( v ) for each pair of vertices u, v ∈ n with u ≥ v , and a -morphism F u,v,v (cid:48) ,w : F ( ϕ v,w ) ◦ F ( ϕ u,v ) −→ F ( ϕ v (cid:48) ,w ) ◦ F ( ϕ u,v (cid:48) ) for each -dimensional face of n with vertices u, v, v (cid:48) , w satisfying u ≥ v, v (cid:48) ≥ w . (1) F − u,v,v (cid:48) ,w = F u,v (cid:48) ,v,w (2) For every -dimensional sub-cube of n , the hexagon of Figure 1 commutes. v (cid:48) wu vw (cid:48) zv (cid:48)(cid:48) w (cid:48)(cid:48) ◦ ◦◦ ◦◦ ◦ F v (cid:48) ,w,w (cid:48) ,z × id id × F u , v (cid:48) , v (cid:48)(cid:48) , w (cid:48) id × F u,v,v (cid:48) ,w F v,w,w (cid:48)(cid:48) ,z × id id × F u , v , v (cid:48)(cid:48) , w (cid:48)(cid:48) F v (cid:48)(cid:48) ,w (cid:48)(cid:48) ,w (cid:48) ,z × id F IGURE -morphisms alongthe faces of -dimensional sub-cubes of n , cf. [AKW, Figure 24].Using Lemma 4.1, the following data gives the oriented Khovanov Burnside func-tor F or : 2 n −→ B σ . F ( u ) is the finite set B ( u ) , defined as the collection of generators B ( W ( u ) , f can ) for the web space HOM
Foam ( ∅ , W ( u )) at the vertex u , constructed inDefinition 3.7. For each edge u ≥ v in n , and y ∈ F or ( v ) , consider the value of theedge differential applied to y : (cid:80) x ∈ F or ( u ) (cid:15) x,y x . By Proposition 3.13, each coefficient (cid:15) x,y is an element of {− , , } . Define F or ( φ u,v ) = { ( y, x ) ∈ F or ( v ) × F or ( u ) | (cid:15) x,y = ± } , where the source and target maps are the projections, and the sign on elements of F or ( φ u,v ) is given by (cid:15) x,y . Ladybug configurations and the completion of the construciton.
The Kho-vanov homotopy type of [LS14a; LLS20] carries more information than the Kho-vanov chain complex, and the additional ingredient in the construction is the anal-ysis of ladybug configurations [LS14a, Section 5.4]. In the setting of the Khovanovhomotopy type there is a unique choice of the -morphisms F u,v,v (cid:48) ,w for square facesin Lemma 4.1, except for the ladybug configuration case.As indicated in Figure 2, such a configuration consists of a simple closed curveand two surgery arcs with endpoints linked on the circle, so that either surgery is asplit into two circles, and the result of both surgeries is again a single component.In the Khovanov chain complex this corresponds to the generator being sent byboth edge maps (comultiplications) to ⊗ X + X ⊗ ; then the multiplication maptakes the result to X . Defining the -morphism F u,v,v (cid:48) ,w in this case amounts toestablishing a bijection between the summands ⊗ X, X ⊗ in the two -componentresolutions; this has to be done consistently so that the hexagon relation in Figure1 holds for each -dimensional cube. This is equivalent to specifying a bijectionbetween the components of these two resolutions; one such correspondence – the“right pair” in the terminology of [LS14a] – is shown in Figure 2. (The labels a, b chosen for illustration here differ from those in [LS14a] to avoid confusion with thelabels , that are used in the gl web context.) The analysis of -dimensional cubeconfigurations in [LS14a, Section 5.5], reformulated in the language of the Burnsidecategory, is used to prove the hexagon relation in [LLS20] when a consistent (rightor left) choice is made for all ladybug configurations. a b a b F IGURE
2. Ladybug configurationIn the Blanchet setting there is still a unique choice of -morphisms F u,v,v (cid:48) ,w , exceptfor the web version of ladybug configurations. Such a configuration is a web W with two zip/unzip moves so that the combinatorics of c ( W ) (the -labeled curvesof W , in the terminology of Section 2) matches that in Figure 2. (Note that arbitrary -labeled edges may be part of the configuration, and are not pictured.)Proposition 3.13 gives the sign coherence of maps in a ladybug configuration. Thevalue of the composition of the two corresponding edge maps in the cube of res-olutions on the undotted generator x is ± times the dotted generator z . (Heredotted and undotted generators are understood in the context of Definition 3.7.)The associated composition of signed correspondences has two elements in the set s − A ( x ) ∩ t − A ( z ) of the same sign, using the notation from (10). As in the setting ofthe Khovanov homotopy type, a consistent choice (say, right pair) is made, ensuring l FOAMS AND THE KHOVANOV HOMOTOPY TYPE 17 that the hexagon in Lemma 4.1 commutes. More specifically, the analysis in [LLS17,Proposition 6.1] shows that the composition of the six -morphisms correspondingto traversing the six faces of a -dimensional cube induces the identity map on the -element correspondence in the Burnside category. In our setting the correspon-dences have signs, which are determined by the signs of the Blanchet differential.However the combinatorics of the ladybug configurations is based on the -labeledcurves; moreover the square faces in the Blanchet complex commute, with no signcorrections. Thus the composition of the six -morphisms is again the identity, veri-fying the hexagon relation.With a functor to the signed Burnside category given by Lemma 4.1, the rest of theconstruction – little boxes refinement (with reflections to accommodate signs) andspacial realization – follows as in [SSS20]. Remark 4.2.
The coefficients of the edge maps with respect to the chosen generatorsare elements of { , ± } ; in this respect our context is similar to that of [SSS20], neces-sitating the use of signed Burnside categories. However, unlike the setting of the oddKhovanov homology in [SSS20], the -dimensional faces of the cube of resolutionsin our context commute (with no sign correction required).Note also that similarly to the odd Khovanov homology context, the differentialsin the Blanchet theory have signs but no cancellations (where a generator is sent byan edge map to a linear combination of generators which then cancel when acted onby another edge map). The presence of cancellations leads to a substantially moreinvolved structure of the moduli spaces in the framed flow category, cf. [AKW20,Section 4]. Remark 4.3.
It is pointed out in [LLS20, Remark 4.22] that incorporating the Bar-Natan and Lee deformations in the construction of the Khovanov homotopy typepresents a problem. The results of Section 3 hold for the deformed foam categories,see Remark 3.16, however working in this setting does not appear to help in con-structing a stable homotopy refinement of the deformed theories. The problem withthe Lee deformation for gl foams is identical to that in [LLS20, Figure 4.1]. For theBar-Natan deformation, the complication is due to the presence of cancellations (dis-cussed in the preceding remark). There is a bit of flexibility in choosing the locationof dots in the generators of the Blanchet theory; moving them across -labeled sheetsresults in a sign change, as shown in equation (4). Nevertheless, it seems unlikelythat there is a consistent choice of generators eliminating cancellations.4.3. Reidemeister moves.
In this section, we prove the homotopy invariance of X or ( L ) under Reidemeister moves. The key facts we need to know about Blanchet’sversion of Khovanov homology, to lift its Reidemeister invariance properties to thelevel of the stable homotopy type X or , are collected in the following proposition. Proposition 4.4.
Let X denote one of the Reidemeister moves from Figure 3, between twooriented link diagrams D and D , where D has more crossings than D . Let r X : CKh or ( D ) −→ CKh or ( D ) denote the associated chain map. Then r X is a deformation retract which factors ↔ , ↔ , ↔ F IGURE
3. Reidemeister moves as r X = φ ◦ r k ◦ · · · ◦ r , where k ∈ N and • each r i for ≤ i ≤ k is a deformation retract induced by “Gaussian elimination”of an acyclic sub- or quotient complex that is spanned by a pair of standard basiselements of CKh or ( D ) if i = 1 , respectively by the images of such a pair under r i − ◦ · · · ◦ r if i > . • there exists a bijection σ between the images of standard basis elements b of CKh or ( D ) ,which are not cancelled in this process, and the standard basis elements of CKh or ( D ) ,such that the foams b and σ ( b ) (Definition 3.7) are isotopic upon forgetting 2-labelledfacets and overall minus signs. • φ is an isomorphism of chain complexes, sending the image of standard basis elements b in r k ◦ · · · ◦ r (CKh or ( D )) to ± σ ( b ) in CKh or ( D ) .Proof. This follows from the discussion of Reidemeister moves in [LS14a] (buildingon [Bar05]) combined with the fact that Blanchet foams reduce to Bar-Natan cobor-disms under the specialisation of coeffiecients to F , and the desired Gaussian elim-inations can be carried out over Z whenever they are possible over F .More specifically, to construct r , we are looking for a pair of generators that forman acyclic sub- or quotient complex. The differentials in the chain complexes CKh or send standard basis elements to linear combinations of standard basis elements withcoefficients from the set {− , , } . In particular, a component of the differential is in-vertible if and only if its specialisation to F coefficients is invertible. Under this spe-cialisation, foams agree with Bar-Natan cobordisms, and thus the discussion from[LS14a] guarantees the existence of a suitable pair of generators, and thus the exis-tence of r . (This pair of generators is local, it exists and has a uniform descriptionwhenever the Reidemeister move of type X is applied within a link diagram.)Now we observe that Gaussian elimination just cancels this pair of generators.Because the pair formed an acyclic sub- or quotient complex, the first intermediatecomplex C , i.e. the co-domain of r , consists of all remaining generators with un-changed differentials between them. In particular, C inherits (not all) standard basiselements and a differential with coefficients from the set {− , , } . Now we iteratethis procedure.The final result r k ◦ · · · ◦ r (CKh or ( D )) is reached, when the total rank of the chaingroups equals that of CKh or ( D ) . In that case, comparison with Bar-Natan cobor-disms via F coefficients establishes the existence of the bijection σ . The isomor-phism φ is then determined by (a single relevant component of) the local model forthe Reidemeister chain map r X as defined by Blanchet. (cid:3) l FOAMS AND THE KHOVANOV HOMOTOPY TYPE 19
The proof that Reidemeister moves induce homotopy equivalences of X or ( L ) isanalogous to the proof of [LLS17, Theorem 1] and [SSS20, Section 5.3]. The deforma-tion retraction r X in Proposition 4.4 gives a natural transformation η : 2 × n −→ B σ with the sign map of F ( ϕ , × Id v ) determined by the isomorphism φ in Proposition4.4. As in the references above, the natural transformation η is a stable equivalenceof functors, giving rise to a homotopy equivalence of spectra associated to link dia-grams related by Reidemeister moves.4.4. Maps induced by cobordisms.
The remaining work concerns establishing nat-urality of the stable homotopy type X or with respect to link cobordisms. As men-tioned in the introduction, [LS14b] defined maps on the Khovanov homotopy typeinduced by a link cobordism represented as a sequence of Reidemeister moves andMorse moves. The well-definedness of this induced map has not yet been verified,but it is conjectured to hold. The construction of the cobordism maps in [LS14b, Sec-tion 3.3] is given in the context of framed flow categories; its analogue in the (signed)Burnside category framework is discussed in [SSS20, Section 5.5]. The result in oursetting reads as follows. Lemma 4.5.
An oriented link cobordism C from L to L (cid:48) , presented as a sequence of Reide-meister and Morse moves, induces a map of spectra X or ( L (cid:48) ) −→ X or ( L ) . The induced mapon cohomology is the map Kh or ( C ) : Kh or ( L ) −→ Kh or ( L (cid:48) ) in Theorem 2.5.Proof. Following the approach of [SSS20], we associate to elementary cobordismsnatural transformations of oriented Burnside functors. This is a lift to the signedBurnside category of the map of functors F Kh of [LLS20].First consider a saddle cobordism between two n -crossing link diagrams, L −→ L (cid:48) .For the Khovanov homotopy type [LS14b] the natural transformation × n −→ B is given by the functor F Kh associated to the link diagram L (cid:48)(cid:48) with an extra cross-ing, corresponding to the saddle. In the Blanchet theory the edge differentials arezip/unzip foams (see Section 2.1) rather than saddles.Consider the ( n + 1) -dimensional cubical diagram of webs and foams, correspond-ing to the cone on the saddle map. Two n -dimensional sub-cubes are given by thelink diagrams L, L (cid:48) , and the additional edge direction corresponds to the saddlecobordism. While this is not a cube associated to a link diagram, it has propertiesanalogous to one. Indeed, it follows from Proposition 3.13 that this cube does nothave cancellations, cf. Remark 4.2. Moreover, the definition of ladybug configura-tions in Section 4.2 is based on -labelled curves of a web, so saddles and zip/unzipfoams are treated equally in ladybug analysis. Therefore the functor × n −→ B σ associated to a saddle cobordism may be defined as in Sections 4.1, 4.2.The natural transformation assigned to a cup (birth) is defined by correspondenceswhich label the new trivial circle with the undotted cup generator; the sign of thiscorrespondence is +1 . For the cap (death) the correspondences involve the trivial cir-cle C labeled by the dotted cup generator. In this case the sign of the correspondenceis (cid:15) , determined by the flow on C , see Lemma 3.14. (cid:3) Relation to the original construction of the Khovanov homotopy type.
In thissection we show that X or ( L ) is homotopy equivalent to X ( L ) of [LS14a]. We start bysummarizing the relevant fact relating the two homology theories. Lemma 4.6.
Let L be an oriented link diagram. There is a chain map Φ : CKh or ( L ) −→ CKh( L ) inducing an isomorphism on homology. On each chain group Φ is a diagonal matrix withdiagonal entries ± with respect to the basis in Definition 3.7 for CKh or ( L ) and the canonicalKhovanov basis for CKh( L ) . The proof follows from [Bel+19, Proposition 4.3 and Theorem 4.6]. A concretedescription of Φ is also given by the results of our Section 3. Proposition 4.7.
Let L be an oriented link diagram. Then X or ( L ) is stably homotopy equiv-alent to X ( L ) .Proof. The statement follows from the claim that the Burnside functors
F, F or : 2 n −→ B σ are naturally isomorphic, where F (thought of as a functor n −→ B σ where all signsare +1 ) was constructed in [LLS20] and F or is the result of Section 4.1. Here a naturalisomorphism is a natural transformation F or −→ F such that for each vertex u themorphism F or ( u ) −→ F ( u ) is an isomorphism in B σ . In the case at hand the naturalisomorphism η arises from the bijection between the basis in Definition 3.7 and thecanonical Khovanov basis; denote this bijection ψ . More precisely, η : 2 × n −→ B σ is defined on objects by η (0 , u ) = F ( u ) , η (1 , u ) = F or ( u ) for u ∈ { , } n . Define η onthe edges e u : (1 , u ) → (0 , u ) to be the signed correspondence η ( e u ) = (cid:16) F or ( u ) id ←− F or ( u ) ψ u −→ F ( u ) (cid:17) , with the sign given by Lemma 4.6. The fact that this indeed gives rise to a naturalisomorphism can be seen for example by following the detailed discussion in [AKW,Section 3.6]. An isomorphism of Burnside functors induces a stable homotopy equiv-alence as stated in the proposition, cf. [SSS20, Lemma 4.17] (cid:3) R EFERENCES [AKW] Rostislav Akhmechet, Vyacheslav Krushkal, and Michael Willis. “Stablehomotopy refinement of quantum annular homology”.
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