AA Bar-Natan homotopy type
Taketo SanoFebruary 16, 2021
Abstract
Following Lipshitz-Sarkar’s construction of Khovanov homotopy type,we construct for any link diagram L a CW spectrum X BN ( L ) whose re-duced cellular cochain complex gives the Bar-Natan complex of L . Weprove that X BN ( L ) is stably homotopy equivalent to the wedge sum of itscanonical cells. Contents
Khovanov [Kho00] introduced
Khovanov homology as a categorification of theJones polynomial. Lipshitz-Sarkar [LS14a] introduced a spatial refinement ofKhovanov homology, called
Khovanov homotopy type , based on the procedureproposed by Cohen-Jones-Segal [CJS95]. The reduced cohomology of Khovanovhomotopy type recovers Khovanov homology, and as a link invariant it is strictlystronger than the original homology theory [See12].1 a r X i v : . [ m a t h . G T ] F e b n the algebraic level, there are two well known deformed versions of Kho-vanov homology, namely Lee homology [Lee05] and
Bar-Natan homology [Bar05].A question arises naturally:
Are there spatial refinements for the deformed the-ories?
The deformations are important in that they give rise to the s -invariant ,an integer valued knot invariant originally introduced by Rasmussen [Ras10]used to reprove the Milnor conjecture [Mil68]. Recently Piccirillo [Pic20] used s to disprove the sliceness of the Conway knot, a problem that remained unsolvedfor half a century. Despite the strength of s , little is known about its geometricorigin. Thus another question arises: Can s be refined homotopically? In this paper we give an answer to the first question, especially for Bar-Natan’s theory.
Theorem 1.
For any link diagram L , there is a CW spectrum X BN ( L ) whosereduced cellular cochain complex gives the Bar-Natan complex C BN ( L ) . In contrast to Khovanov homology, Bar-Natan homology is characteristic inthat its graded module structure is completely known: it is freely generated bythe canonical classes of L . This fact also lifts to the spatial level: Theorem 2. X BN ( L ) is stably homotopy equivalent to the wedge sum the canon-ical cells. In particular, the stable homotopy type of X BN ( L ) is a link invariant. As a corollary, we obtain properties whose counterparts in Khovanov homo-topy type are known to be true [LLS20, Theorem 1, 11]:
Corollary 1.1.
For link diagrams
L, L (cid:48) ,1. X BN ( L (cid:116) L (cid:48) ) (cid:39) X BN ( L ) ∧ X BN ( L (cid:48) ) .2. X BN ( m ( L )) (cid:39) X BN ( L ) ∨ .Here m ( · ) denotes the mirror, and ( · ) ∨ denotes the Spanier-Whitehead dual. The s -invariant is defined using the quantum filtration on Bar-Natan ho-mology ([Ras10], [MTV07], [LS14b]). It is natural to expect that the quantumfiltration also lifts to the spatial level. To pursue this, there is a difficulty indealing with the high dimensional moduli spaces of the framed flow category.We conjecture the following two: Conjecture 1.
There is an ascending filtration on the spectrum X = X BN ( L ) { pt } = F m X ⊂ · · · ⊂ F j X ⊂ F j +2 X ⊂ · · · ⊂ F M X = X that respects the quantum filtration on the Bar-Natan complex. Conjecture 2.
For each j , the quotient F j X /F j − X is stably homotopy equiv-alent to the j -th wedge summand X j Kh ( L ) of the Khovanov spectrum X Kh ( L ) . Examples are given for which the conjectures are true. We expect thatproving the conjectures will pave a way to homotopically refining the s -invariant.2 dea and method As pointed out in [LLS20, Remark 4.22], there are difficulties in applying theconstruction of Khovanov homotopy type to other deformations. Deformedtheories are given by replacing the defining Frobenius algebra of Khovanov ho-mology A = R [ X ] / ( X ) with A h,t = R [ X ] / ( X − hX − t ). In order to apply theconstruction, the following conditions are necessary: (i) the coefficients appear-ing in the multiplication and the comultiplication of A h,t are all 1 (with respectto the standard basis { , X } ), and (ii) the poset associated to the chain complexis cubic in a certain sense. Both Lee’s theory ( h, t ) = (0 ,
1) and Bar-Natan’stheory ( h, t ) = (1 ,
0) do not satisfy the two conditions.However, for Bar-Natan’s theory, by taking basis { X, Y = X − } insteadof { , X } , the multiplication and the comultiplication of A , diagonalizes as: m ( X ⊗ X ) = X, m ( X ⊗ Y ) = m ( Y ⊗ X ) = 0 , m ( Y ⊗ Y ) = − Y, ∆( X ) = X ⊗ X, ∆( Y ) = Y ⊗ Y. Then the first condition is almost satisfied. That the second condition holdscan be proved rather easily. Thus by taking extra care of the signs, we showthat the original construction of Khovanov homotopy type can be applied toBar-Natan’s theory.Throughout this paper, we use a set of techniques called the flow categorymoves developed by Lobb et al. in [JLS17] and in [LOS18]. Analogous to themoves in classical Morse theory, these moves allow one to modify a framedflow category without changing the stable homotopy type of the correspondingspectrum. For our situation, the basis { , X } is recovered from { X, Y } on theflow category level by performing handle slides . Plan of the paper
In Section 2, we review the basics of Khovanov homology and framed flow cat-egories. We also summarize the three flow category moves given in [JLS17] andin [LOS18]. In Section 3, we give the construction of the XY -based Bar-Natanflow category C BN ( L ) and endow it an appropriate framing. Then the associ-ated spectrum X BN ( L ) gives the spatial refinement of the Bar-Natan complex C BN ( L ) (Theorem 1). We also determine the stable homotopy type of X BN ( L )as the wedge sum of the canonical cells (Theorem 2). In Section 4, we apply han-dle slides to C BN ( L ) to obtain the 1 X -based Bar-Natan flow category C (cid:48) BN ( L ).This allows us to define the quantum gradings on the objects of C (cid:48) BN ( L ). Fi-nally we state the two conjectures (Conjecture 1, 2) together with examples thatsupport the conjectures. 3 Preliminaries
Throughout this paper we assume links and link diagrams are smooth and ori-ented.
Definition 2.1 (Frobenius algebras) . Let R be a commutative ring with unity.A Frobenius algebra over R is a quintuple ( A, m, ι, ∆ , ε ) such that:1. ( A, m, ι ) is an associative R -algebra with multiplication m and unit ι ,2. ( A, ∆ , ε ) is a coassociative R -coalgebra with comultiplication ∆ and counit ε ,3. the Frobenius relation holds:∆ ◦ m = ( id ⊗ m ) ◦ (∆ ⊗ id ) = ( m ⊗ id ) ◦ ( id ⊗ ∆) . Definition 2.2.
For any h, t ∈ R , let A h,t = R [ X ] / ( X − hX − t ). A h,t given aFrobenius algebra structure as follows: First, the R -algebra structure is obvious.The counit ε : A h,t → R is defined by ε (1) = 0 , ε ( X ) = 1 . Then the comultiplication ∆ is uniquely determined so that ( A h,t , m, ι, ∆ , ε )becomes a Frobenius algebra. Explicitly, m and ∆ are given by m (1 ⊗
1) = 1 , m ( X ⊗
1) = m (1 ⊗ X ) = X, m ( X ⊗ X ) = hX + t, ∆(1) = X ⊗ ⊗ X − h (1 ⊗ , ∆( X ) = X ⊗ X + t (1 ⊗ . (2.1) Definition 2.3 (Sign assignments) . Let K ( n ) be the n -dimensional unit cubeendowed with the standard CW-complex structure. A 1-cochain s ∈ C ( K ( n ); F )is called a sign assignment for K ( n ) if δs ( e ) = 1 for every 2-cell e of K ( n ).The standard sign assignment s is defined as s ( e ) = ε · · · ε i − for an edge e = ( ε , . . . , ε i − , (cid:63), ε i +1 , . . . , ε n ) . Definition 2.4 (Khovanov complex and homology) . Fix any h, t ∈ R . Givena link diagram L with n crossings and a sign assignment s for K ( n ), the (gen-eralized) Khovanov complex C h,t ( L, s ) is defined by the construction given in[Kho00], except that the defining Frobenius algebra A = R [ X ] / ( X ) is replacedby A h,t = R [ X ] / ( X − hX − t ). The homology of C h,t ( L, s ) is called the (gen-eralized) Khovanov homology of L and is denoted H h,t ( L, s ). Definition 2.5 (States, enhanced states) . A state u of L is an assignment of 0or 1 to each crossing of L . When a total ordering of the crossings of L is given,then u is identified with an element of { , } n . A state u yields a crossinglessdiagram D L ( u ) by resolving all crossings accordingly. For any basis S for A h,t ,an S -enhanced state of L is a pair x = ( u, x ) such that u is a state and x is anassignment of an element in S to each circle of D L ( u ). An enhanced state x isregarded as an element of C h,t ( L, s ) and all together form a basis of C h,t ( L, s ).4 efinition 2.6 (Homological grading and quantum grading) . Let n ± denotethe number of positive / negative crossings of L . The homological grading of anenhanced state x = ( u, x ) of L is given bygr h ( x ) = | u | − n − , where | u | is the number of 1’s in u . Next, take the basis S = { , X } for A h,t .For any 1 X -enhanced state x , the quantum grading of x = ( u, x ) is defined asfollows: declare deg(1) = 1 , deg( X ) = − x ) be the sum of thedegrees of the labels on the circles of D L ( u ). Definegr q ( x ) := gr h ( x ) + deg( x ) + n + − n − . It easily follows that gr q ( x ) ≡ | L | (mod 2) ([Kho00, Proposition 24.]). When( h, t ) = (0 , d of C h,t ( L, s ) preserves the quantum grading andhence the homology inherits the bigrading. Otherwise, unless R is graded, d isquantum grading non-decreasing. Thus the chain complex and the homologyadmit filtrations. It is proved in [Kho06] that the isomorphism class (as abigraded module or a filtered graded module) of H h,t ( L, s ) is independent of s and is invariant under the Reidemeister moves. Thus it is justified to refer tothe isomorphism class as the Khovanov homology of the corresponding link.The original theory H Kh is given by the specialization ( h, t ) = (0 , H Lee is given by ( h, t ) = (0 ,
1) and Bar-Natan’s theory H BN is givenby ( h, t ) = (1 , H Lee ( − ; Q ) is generated by the canonical generators and always has dimension 2 | L | . This is in contrast to H Kh ,where the complexity of the module structure increases with crossing number ofthe link. The structure theorem for Q -Lee homology generalizes to H h,t underthe following condition. Condition 2.7 (Diagonalizable condition) . h, t ∈ R satisfy the following con-dition: the quadratic X − hX − t factors as ( X − u )( X − v ) in R [ X ] and thedifference of the two roots c := v − u is invertible in R .Assume that Condition 2.7 holds. If we take the { a = X − u, b = X − v } for A = A h,t , the multiplication and the comultiplication diagonalizes as: m ( a ⊗ a ) = c a , m ( a ⊗ b ) = m ( b ⊗ a ) = 0 , m ( b ⊗ b ) = − c b , ∆( a ) = a ⊗ a , ∆( b ) = b ⊗ b . (2.2) Algorithm 2.8 ( ab -labeling of Seifert circles) . Given a link diagram L , welabel each of its Seifert circles by a or b according to the following algorithm:separate R into regions by the Seifert circles of L , and color the regions inthe checkerboard fashion (with the unbounded region colored white). For eachSeifert circle, let it inherit the orientation from L , and label a if it sees a blackregion to the left, otherwise label b (see Figure 1). Definition 2.9 (Canonical cycles) . Let L be a link diagram. There are 2 | L | possible orientations on (the underlying unoriented diagram of) L . For each5 b Figure 1: The canonical cyclesorientation o , there is a unique state u such that resolving L by u give theSeifert circles of L reoriented with o . By regarding ab -labeled circles as an ab -enhanced state, Algorithm 2.8 determines an element α ( L, o ) in C h,t ( L, s ). Onecan easily prove that these elements are cycles. They are called the canonicalcycles of L , and those homology classes are called the canonical classes of L . Proposition 2.10 ([Lee05, Theorem 4.2], [Tur20, Theorem 4.2], [San20, Propo-sition 2.9]) . When Condition 2.7 is satisfied, H h,t ( L ) is freely generated by thecanonical classes { [ α ( L, o )] } o . In particular it is a free R -module of rank | L | . Proposition 2.11 ([Lee05, Proposition 4.3]) . Let L be an (cid:96) -component linkdiagram and L , · · · , L l be the diagrams corresponding to the components. Forany orientation o on (the underlying unoriented diagram of ) L , let I ⊂ { , . . . , l } be the set of indices i such that o is opposite on L i to the given orientation. Thehomological grading of α ( L, o ) is given by: (cid:88) i ∈ I,j / ∈ I lk ( L i , L j ) . where lk denotes the linking number. In particular, when o is the given orien-tation of L , then gr( α ( L, o )) = 0 . From Proposition 2.10 and Proposition 2.11, when Condition 2.7 is satisfied,the graded module structure of H h,t ( L ) is completely determined. Note that Q -Lee homology satisfies Condition 2.7 for X − X − X + 1) and c = 2is invertible in Q . For Bar-Natan homology, we have X − X = X ( X − c = 1, so Condition 2.7 is satisfied over Z (or any other R ). This is whatmakes the construction of Bar-Natan homotopy type possible. For Bar-Natanhomology, we use letters X and Y = X − a and b . Definition 2.12 (Flow categories) . A flow category C is a topological categoryequipped with a grading function | · | : Ob( C ) → Z that satisfies the following conditions 6. For any x ∈ C , Hom C ( x, x ) = { id x } . For any distinct x, y ∈ C , Hom C ( x, y ) = M C ( x, y ), where M C ( x, y ) is a (possibly empty) compact ( | x | − | y | − (cid:104)| x | − | y | − (cid:105) -manifold . M C ( x, y ) is called the moduli space from x to y .2. For any distinct x, y, z ∈ C with | z | − | y | = k , the composition map: ◦ : M C ( z, y ) × M C ( x, z ) → M C ( x, y )is a (cid:104)| x | − | y | − (cid:105) -map, i.e. ◦ − ( ∂ i M C ( x, y )) = (cid:40) ∂ i M C ( z, y ) × M C ( x, z ) for i < k, M C ( z, y ) × ∂ i − k +1 M C ( x, z ) for i ≥ k.
3. For any distinct x, y ∈ C , the composition map ◦ induces a diffeomorphism ∂ i M C ( x, y ) ∼ = (cid:71) z : | z |−| y | = i M C ( z, y ) × M C ( x, z ) . Definition 2.13 (Framed flow categories) . A framed flow category is a triple( C , ι, ϕ ) such that C is a flow category, ι is a neat embedding of C into someEuclidean space with corners, and ϕ is a coherent framing for ι (see [LS14a] forthe precise definition). We usually make ι and ϕ implicit and refer to C as aframed flow category. Definition 2.14 (Associated (co)chain complex) . Given a framed flow category C = ( C , ι, ϕ ), the associated chain complex C ∗ ( C ) of C is defined as follows:1. The i -th chain group is freely generated by the grading i objects of C : C i ( C ) = (cid:77) | x | = i Z x.
2. The differential ∂ is defined as ∂x = (cid:88) | y | = | x |− M C ( x, y ) y Here, for each point p ∈ M C ( x, y ), its sign is given by the orientation ofthe framing ϕ ι x,y ( p ) at the point p .The associated cochain complex C ∗ ( C ) is the dual complex Hom( C ∗ ( C ) , Z ).Given a (co)chain complex C , we say C refines C if the associated (co)chaincomplex of C is isomorphic to C . An (cid:104) n (cid:105) -manifold M is a manifold with corners, together with faces ( ∂ M, . . . , ∂ n M ) thatgive a combinatorial structure on the boundary ∂M = (cid:83) i ∂ i M . See [LS14a] for the precisedefinition. roposition 2.15 ([LS14a, Section 3.3]) . Given a framed flow category C , thereis a CW complex |C| such that (i) the cells of |C| correspond bijectively to theobjects of C , (ii) there is an integer (cid:96) ∈ Z such that the cell σ x correspondingto an object x ∈ Ob( C ) has dim( σ x ) = | x | + (cid:96) , and (iii) there is a canonicalisomorphism: (cid:101) C ∗ ( |C| )[ − (cid:96) ] ∼ = C ∗ ( C ) where (cid:101) C ∗ denotes the reduced cellular chain complex and [ · ] denotes the gradingshift. Moreover, the formal desuspension Σ − (cid:96) |C| is independent of the auxiliarychoices made for the construction of |C| , up to stable homotopy equivalence. Definition 2.16 (Associated spectrum) . The CW complex |C| is called the realization of the framed flow category C , and the spectrum X ( C ) := Σ − (cid:96) |C| iscalled the associated spectrum of C .The following example will be necessary in the constructions of both Kho-vanov homotopy type and Bar-Natan homotopy type. Definition 2.17 (Cube complex) . Given any sign assignment s for K ( n ), the n -dimensional cube complex C ∗ ( n, s ) is the chain complex freely generated by { , } n and the differential ∂ given by ∂u = (cid:80) u> v ( − s ( e ( u,v )) v , where e ( u, v )is the edge of K ( n ) that connects u and v . Definition 2.18 (Cube flow category) . The n -dimensional cube flow category C cube ( n ) is defined as follows: the object set is { , } n , and the moduli spacebetween vertices u > k v is the ( k − permutohedron Π k − . Thecomposition map is defined to satisfy some coherence condition (see [LLS20,Definition 3.16] for the precise definition). Remark . Originally in [LS14a, Definition 4.1], C cube ( n ) is defined as theMorse flow category of the following Morse function: f n : R n → R , ( x , . . . , x n ) (cid:55)→ f ( x ) + · · · + f ( x n )where f : R → R is a self-indexing Morse function with critical points { , } .The object set of C cube ( n ) is the critical points of f n , and the moduli spacebetween ciritical points u > v is the moduli space of gradient flow lines thatflows from u to v . Proposition 2.20 ([LS14a, Proposition 4.12]) . Given any sign assignment s for K ( n ) , there is a framed neat embedding ( ι, ϕ ) of C cube ( n ) such that the framedflow category ( C cube ( n ) , ι, ϕ ) refines the cube complex C ∗ ( n, s ) . Here we summarize the three flow category moves given in [JLS17] and in[LOS18]. See Figure 2, 3, 4 for schematic diagrams of the following three propo-sitions. 8 roposition 2.21 (Handle cancellation, [JLS17, Theorem 2.17]) . Let C be aframed flow category. Suppose there are objects x, y ∈ C such that | x | = | y | + 1 and the -dimensional moduli space M C ( x, y ) consists of a single point. Thenthere is a framed flow category C (cid:48) satisfying the following:1. Ob( C (cid:48) ) = Ob( C ) \ { x, y } .2. For any objects a, b ∈ C (cid:48) , M C (cid:48) ( a, b ) = ( M C ( a, b ) (cid:116) M C ( x, b ) × M C ( a, y )) / ∼ where ∼ identifies boundaries M C ( x, b ) × M C ( a, x ) ∪ M C ( y, b ) × M C ( a, y ) and M C ( x, b ) × ( M C ( a, x ) × M C ( a, x )) ∪ ( M C ( y, b ) × M C ( x, y )) × M C ( a, y ) . |C (cid:48) | is homotopy equivalent to |C| . Proposition 2.22 (Handle sliding, [LOS18, Theorem 3.1]) . Let C be a framedflow category. For any choices of a sign ε ∈ {±} and objects x, y ∈ C of thesame grading, there is a framed flow category C (cid:48) satisfying the following:1. Ob( C (cid:48) ) = (Ob( C ) \ { x } ) ∪ { x (cid:48) } .2. For any objects a, b ∈ C (cid:48) , M C (cid:48) ( a, y ) = ( − ε ) M C ( a, x ) (cid:116) M C ( a, y ) , M C (cid:48) ( x (cid:48) , b ) = M C ( x, b ) (cid:116) ( ε ) M C ( y, b ) , M C (cid:48) ( a, b ) = M C ( a, b ) (cid:116) ( M C ( y, b ) × [0 , × M C ( a, x )) . Here, the signs ( ± ε ) indicate that the framings are changed accordingly.3. |C (cid:48) | is homotopy equivalent to |C| . The object x (cid:48) ∈ C (cid:48) is said to be obtained by sliding x over y . Proposition 2.23 (Whitney trick, [JLS17, Theorem 2.8]) . Let C be a framedflow category. Suppose there are objects x, y ∈ C such that | x | = | y | + 1 and the -dimensional moduli space M C ( x, y ) contains two points P, Q with oppositeframings. Then there is a framed flow category C (cid:48) satisfying the following:1. Ob( C (cid:48) ) = Ob( C ) .2. For the two objects x, y , M C (cid:48) ( x, y ) = M C ( x, y ) \ { P, Q } . or any objects a, b , M C (cid:48) ( a, y ) = ( M C ( a, y ) \ { P, Q } × M C ( a, x )) / ∼ , M C (cid:48) ( x, b ) = ( M C ( x, b ) \ M C ( y, b ) × { P, Q } ) / ∼ , M C (cid:48) ( a, b ) = ( M C ( a, b ) \ M C ( y, b ) × { P, Q } × M C ( a, x )) / ∼ . Here, the identifications ∼ are defined so that the corners of the modulispaces are glues appropriately (see [JLS17, Definition 2.5] for the precisedefinition).3. |C (cid:48) | is homotopy equivalent to |C| .Remark . In [LOS18, Section 4], a more generalized extended Whitney trick is introduced, where x, y need not satisfy | x | = | y | + 1, and the moduli space M C ( x, y ) can be replaced by any framed cobordant manifold. Definition 2.25 (Move equivalence) . Two framed flow categories are moveequivalent if they can be interchanged by a finite sequence of flow categorymoves (plus isotopies and stabilizations of the framed neat embeddings). Wealso call the stable homotopy equivalence between the two corresponding spectraa move equivalence , and say that the two spectra are move equivalent . Remark . It is conjectured in [JLS17, Conjecture 6.4] that two framed flowcategories determine the same stable homotopy type if and only if they are moveequivalent.
Definition 3.1 (Resolution configurations) . A resolution configuration is a pair D = ( Z ( D ) , A ( D )), where Z ( D ) is a set of pairwise-disjoint embedded circlesin S , and A ( D ) is a totally ordered collection of disjoint arcs embedded in S satisfying A ( D ) ∩ Z ( D ) = ∂A ( D ). The cardinality of A ( D ) is called the index of D and is denoted ind ( D ). D is basic if every circle of Z ( D ) intersects an arcin A ( D ). D is connected if the underlying planar graph of D is connected. Definition 3.2 (Surgery) . Let D be an index n resolution configuration. Forany subset B ⊂ A ( D ), the surgery of D along B , denoted s B ( D ), is the reso-lution configuration obtained by performing embedded surgeries along all arcs a ∈ B . In particular the maximal surgery on D is denoted s ( D ) := s A ( D ) ( D ).When E = s B ( D ) with | B | = k , we write E (cid:23) k D . Since A ( D ) is totallyordered, a subset B ⊂ A ( D ) can be identified with a vertex u ∈ { , } n , so wealso write D ( u ) for s B ( D ). Definition 3.3 (Subtraction) . For resolution configurations
D, E , the subtrac-tion D \ E is the resolution configuration with circles Z ( D ) \ Z ( E ) and arcs { a ∈ A ( D ) | a ∩ (cid:83) Z ( E ) = ∅ } . 10 • a • x • ◦ y • ◦ b • b • Figure 2: Handle cancellation a • a • x • y • x (cid:48) • y • b • b • α β βα ∓ α ( ± ) γ δ γ ± δ δ Figure 3: Handle slide a • a • x • x • y • y • b • b • + − Figure 4: Whitney trick11igure 5: A resolution configuration associated to a link diagram
Definition 3.4 ( S -labeling) . Let S be a set. An S -labeling of a resolutionconfiguration D is a map x : Z ( D ) → S. Such pair (
D, x ) is called an S -labeled resolution configuration . When Z ( D ) istotally ordered as { Z i } ri =1 , we formally express x as x = x ⊗ · · · ⊗ x r , x i = x ( Z i ) . Definition 3.5 (Basic relations) . Let S be the set of all S -labeled resolutionconfigurations. A basic relation is a binary relation ≺ on S such that ( E, y ) ≺ ( D, x ) holds only if (i) E is basic, (ii) ind( E ) = 1 and (iii) D = s ( E ). A basicrelation ≺ generates a partial order (cid:22) on S as follows: First, extend ≺ as abinary relation on S by declaring ( E, y ) ≺ ( D, x ) if and only if (i) D ≺ E , (ii)the basic relation holds between circles that are involved in the surgery:( E \ D, y | E \ D ) ≺ ( D \ E, x | D \ E )and (iii) the labelings are identical on the circles that are not involved in thesurgery. Then (cid:22) is defined by the reflexive and transitive closure of ≺ . Wewrite ( E, y ) (cid:22) k ( D, x ) when (
E, y ) (cid:22) ( D, x ) and ind( E ) − ind( D ) = k . Definition 3.6 (Triads) . An S -triad is a triple ( D ; x, y ) such that D is aresolution configuration, y is an S -labeling of D , and x is an S -labeling of s ( D ). An S -triad is admissible if ( D, y ) (cid:22) ( s ( D ) , x ) holds. Definition 3.7 (Associated posets) . Suppose S is given a partial order (cid:22) . Forany resolution configuration D , the poset P S, (cid:22) ( D ) is defined by P S, (cid:22) ( D ) := { ( E, x ) | E = s B ( D ) , B ⊂ A ( D ) } . For an S -triad ( D ; x, y ), a (possibly empty) subposet P S, (cid:22) ( D ; x, y ) of P S, (cid:22) ( D )is defined by P S, (cid:22) ( D ; x, y ) := { ( E, z ) ∈ P S, (cid:22) ( D ) | ( D, y ) (cid:22) ( E, z ) (cid:22) ( s ( D ) , x ) } . Obviously P S, (cid:22) ( D ; x, y ) is non-empty if and only if ( D ; x, y ) is admissible. In [LS14a], admissible 1 X -triads are called decorated resolution configurations.
12 1 1 X XX X X X XX Figure 6: The basic relation for X Kh Definition 3.8 (Resolution configurations associated to link diagrams) . Let L be a link diagram with a fixed total ordering of its crossings. The resolutionconfiguration D L associated to L is the resolution configuration obtained byperforming 0-resolutions on all crossings of L , and arcs inserted one for eachcrossing so that it is perpendicular to the resolution.For comparison with the later arguments, we briefly describe the originalconstruction of Khovanov homotopy type given in [LS14a]. The label set is givenby S = { , X } , and the basic relation ≺ is defined as in Figure 6 (compare (2.1)for ( h, t ) = (0 , X -labeled resolution configuration ( D L ( u ) , x )can be identified with a 1 X -enhanced state ( u, x ) of L . For any sign assignment s of K ( n ), the differential d of C Kh ( L, s ) can be written as d y = (cid:88) y ≺ x ( − s ( e ( u,v )) x where u, v ∈ { , } n are the states of corresponding to x , y respectively. The Khovanov flow category C Kh ( L ) is constructed, inductively from the low di-mensional moduli spaces, so that each moduli space M Kh ( x , y ) maps onto the( | x | − | y | − F from C Kh ( L ) to the n -dimensional cube flow category C cube ( n ). From Proposition 2.20, there is a framing of C cube ( n ) relative to s .This is pulled back to C Kh ( L ) by F , and then the framed flow category C Kh ( L, s )refines C Kh ( L, s ). The
Khovanov spectrum X Kh ( L ) is defined as the spectrumassociated to C Kh ( L, s ). Now we construct the refinement C BN ( L, s ) of the Bar-Natan complex C BN ( L ).Recall from Section 2.1 that the operations of the defining Frobenius algebra A = A , is diagonalized by the basis { X, Y = X − } . Take the label set S = { X, Y } and consider XY -labeled resolution configurations, with the basicrelation defined as in Figure 7 (compare (2.2) for c = 1). Note that the negativesign in one of the equation has disappeared in the corresponding relation. Thiswill be taken care lately by the framing.13 X X XXYX YYY Y Y
Figure 7: The basic relation for X BN For any resolution configuration D , we write P BN ( D ) := P XY, (cid:22) ( D ), andfor any triad ( D ; x, y ) we write P BN ( D ; x, y ) := P XY, (cid:22) ( D ). We will see that P BN ( D ) is a disjoint union of cubic posets. Proposition 3.9.
There are natural poset isomorphisms: P BN ( D ; x, y ) ∼ = P BN ( D ∗ ; y, x ) op ,P BN ( D ) ∼ = P BN ( D ∗ ) op . Here D ∗ is the dual resolution configuration of D .Proof. The basic relation is invariant under the reversal of arrows.
Proposition 3.10.
Let D be a resolution configuration with connected compo-nents D , · · · , D (cid:96) . An XY -triad ( D ; x, y ) is admissible if and only if (i) y isconstant on D i , (ii) x is constant on s ( D i ) and (iii) y | D i = x | s ( D i ) for eachcomponent D i .Proof. If there is an arc that connects differently labeled circles, there will beno available labeling after the merge of the two circles.
Proposition 3.11.
Let D be an index k resolution configuration. For anyadmissible XY -triad ( D ; x, y ) , there is a natural poset isomorphism P BN ( D ; x, y ) ∼ = { , } k . Proof.
Take any (
E, z ) ∈ P BN ( D ; x, y ). From Proposition 3.10, z is auto-matically determined from E , and E corresponds one-to-one to a subset B ⊂ A ( D ). Proposition 3.12.
Let D be a resolution configuration. For any ( E, z ) ∈ P BN ( D ) , there is a unique maximal object that succeeds or equals ( E, z ) , and aunique minimal object that precedes or equals ( E, z ) .Proof. Let B be the subset of A ( D ) that contains all of the arcs that connectdifferently labeled circles of Z ( E ). Then the maximal object ( E (cid:48) , z (cid:48) ) is givenby E (cid:48) = s A ( D ) \ B ( D ) with z (cid:48) automatically determined so that ( E, z ) (cid:22) ( E (cid:48) , z (cid:48) ).The latter statement follows from Proposition 3.9.14 efinition 3.13. Let P be a poset and x, y ∈ P . The full subposet between x and y is the subposet { z ∈ P | y ≤ z ≤ x } (empty when y (cid:54)≤ x ). Proposition 3.14.
Let D be a resolution configuration. For ( E, y ) , ( E (cid:48) , x ) ∈ P BN ( D ) , the full subposet between ( E, y ) , ( E (cid:48) , x ) is non-empty if and only if the XY -triad ( E \ E (cid:48) ; x | , y | ) is admissible. In such case, the subposet is isomorphicto P BN ( E \ E (cid:48) ; x | , y | ) . Proposition 3.15.
Let D be a resolution configuration. The poset P BN ( D ) decomposes into a disjoint union of full subposets each containing a unique min-imal element ( D i , y i ) and a unique maximal element ( D (cid:48) i , x i ) . Thus, P BN ( D ) ∼ = (cid:71) i P BN ( D i \ D (cid:48) i ; x i , y i ) ∼ = (cid:71) i { , } k i where k i = ind( D i \ D (cid:48) i ) .Proof. Take the set of all minimal objects { ( D i , y i ) } i in P BN ( D ). From Proposi-tion 3.12, for each i there is a unique maximal object ( D (cid:48) i , x i ) (cid:23) ( D i , y i ). Againfrom Proposition 3.12, these { ( D (cid:48) i , x i ) } are distinct. Obviously any element( E, z ) belongs to exactly one of these full subposets.We call the decomposition of Proposition 3.15 the cubic decomposition of P BN ( D ), and each full subposet a cubic subposet . There is a poset homomor-phism F : P BN ( D ) → { , } n , ( D ( u ) , z ) (cid:55)→ u and is injective on each cubic subposet. We define the flow category ¯ C BN ( D ) sothat F lifts as a functor: ¯ C BN ( D ) C cube ( n ) P BN ( D ) op ( { , } n ) op F F op Definition 3.16.
Let D be an index n resolution configuration D . The flowcategory ¯ C BN ( D ) is defined as follows: when D is non-empty, the object set of¯ C BN ( D ) is given by P BN ( D ), and the grading function is defined bygr ( E, z ) = n − ind( E ) ∈ [0 , n ] . Consider the cubic decomposition of P BN ( D ): P BN ( D ) = (cid:71) i P BN ( D ) i . The moduli spaces of ¯ C BN ( D ) is given by pulling back the moduli spaces of C cube ( n ) by the inclusion F | P BN ( D ) i : P BN ( D ) i (cid:44) → { , } n . F : ¯ C BN ( D ) → C cube ( n )be the functor that lifts F op in the obvious way. When D is empty, define C BN ( ∅ ) = (the discrete category with a single object) and F = id . Remark . This construction is in contrast to that of Khovanov flow cate-gory C Kh , where the moduli spaces are constructed inductively from the lowerdimensions, so that each moduli space of C Kh covers a moduli space of the cubeflow category.Next we endow ¯ C BN ( D ) a framing relative to a sign assignment s for K ( n ).Recall from Proposition 2.20 that there is a framed neat embedding ( ι, ϕ ) of C cube ( n ) such that the associated chain complex of ( C cube ( n ) , ι, ϕ ) gives thecube complex C ∗ ( n, s ). We will not simply pullback ( ι, ϕ ) to ¯ C BN ( D ) by F , forwe need to take care of the negative sign appearing in m ( Y ⊗ Y ) = − Y. Let ( D ; x, y ) be an index k admissible triad. Then P BN ( D ; x, y ) ∼ = { , } k .Let t ∈ C ( K ( k ); F ) be a 1-cochain so that t ( e u,v ) = 1 if and only if thecorresponding surgery D ( v ) ≺ D ( u ) merges two circles labeled Y . Lemma 3.18. t is a cocycle.Proof. Take any 2-face σ of K ( k ). We prove that the total number of merges of Y -labeled circles at the four boundary edges is even. Let a, b the correspondingtwo arcs of D . If a, b belongs to different components of D , the claim is obvious.Otherwise, y is constant on the component and again the claim is obvious. Thus δt ( σ ) = t ( ∂σ ) = 0 ∈ F .Now let D be an index n resolution configuration and s be a sign assignmentfor K ( n ). Consider the cubic decomposition P BN ( D ) ∼ = (cid:71) i P BN ( D ) i where each P BN ( D ) i is the full subposet of P BN ( D ) between ( D ( u i ) , x i ) (cid:23) k i ( D ( v i ) , u i ). For each i , let I u i ,v i : K ( k i ) (cid:44) → K ( n )be the obvious inclusion such that I u i ,v i (¯0) = v i and I u i ,v i (¯1) = u i . Let s i =( I u i ,v i ) ∗ s , and t i be the above defined 1-cocycle on K ( k i ) corresponding to thetriad ( D ( v i ) \ D ( u i ); x i | , y i | ). Then s (cid:48) i = s i + t i is another sign assignment on K ( k i ). Let C BN ( D ) i ∼ = C cube ( k i ) be the subcategory of C BN ( D ) that correspondsto P BN ( D ) i . From Proposition 2.20 there is a framed neat embedding ( ι i , ϕ i )of C BN ( D ) i relative to s (cid:48) i . Assembled together,( ι, ϕ ) = (cid:71) i ( ι i , ϕ i )gives a framed neat embedding of ¯ C BN ( D ) (after some perturbation).16 roposition 3.19. The framed flow category ( ¯ C BN ( D ) , ι, ϕ ) refines the unnor-malized Bar-Natan complex ¯ C BN ( D, s ) .Proof. Let C ∗ be the associated cochain complex of the framed flow category( ¯ C BN ( D ) , ι, ϕ ). Let x ∗ denote the generator of C ∗ dual to the object x . Theisomorphism is given by mapping x ∗ (cid:55)→ x . Indeed, the coboundary map δ of C ∗ is given by δ y ∗ = (cid:88) y ≺ x M ( x , y ) · x ∗ where M ( x , y ) is the signed counting of the 0-dimensional moduli space M ( x , y ), which in this case is a single point. Take any object y , and sup-pose P BN ( D ) i is the subposet that contains y . Take any x (cid:31) y , which alsobelongs to P BN ( D ) i . Then from Proposition 2.20 we have M ( x , y ) = ( − ( s i + t i )(¯ e ) where ¯ e is an edge of K ( k i ) corresponding to y ≺ x in P BN ( D ) i . We have s i (¯ e ) = s ( e )where e = I u i ,v i (¯ e ) is an edge of K ( n ) corresponding to y ≺ x in P BN ( D ). Onthe other hand, we have t i (¯ e ) = 1 if and only if y ≺ x corresponds to a mergeof two circles labeled Y . Thus we have δ y ∗ = (cid:88) y ≺ x ( − s ( e ) ( − t i (¯ e ) x ∗ . Comparing equations (2.2) and the basic relation Figure 7, we see that δ isidentical to the differential d of ¯ C BN ( D, s ). Definition 3.20 (Bar-Natan spectrum) . Let L be a link diagram with a fixedtotal ordering of its crossings. Let n − be the number of the negative crossingsof L . Define the ( XY -based) Bar-Natan flow category by C BN ( L ) := ¯ C BN ( D L )[ − n − ] . For any sign assignment s for K ( n ), let C BN ( L, s ) denote the framed flow cat-egory with the framed neat embedding of C BN ( L ) constructed as above. The ( XY -based) Bar-Natan spectrum of L is defined as the associated spectrum of C BN ( L, s ), and is denoted X BN ( L, s ). Remark . When L is empty, X BN ( ∅ ) = S (the sphere spectrum).Thus we obtain the first theorem: Theorem 1.
There is an isomorphism (cid:101) C ∗ ( X BN ( L, s )) ∼ = −−→ C ∗ BN ( L, s ) that maps the dual cells of X BN ( L, s ) to the XY -enhanced states of C ∗ BN ( L, s ) . aa a b bb b a ba b Figure 8: Local patterns of an admissible edge labeling
On the algebraic level, the graded module structure of H BN ( L ) is completelydetermined: it is freely generated by the canonical classes { [ α ( L, o )] } o . We liftthis statement to the spacial level. Definition 3.22 (Canonical objects, canonical cells) . For each orientation o on (the underlying unoriented link diagram of) L , Algorithm 2.8 determinesan XY -labeled resolution configuration x α ( L, o ) and the corresponding cell σ α ( L, o ) of X BN ( L ). We call { x α ( L, o ) } o the canonical objects of C BN ( L ), and { σ α ( L, o ) } o the canonical cells of X BN ( L ).Obviously, the dual canonical cells σ ∗ α ( L, o ) ∈ (cid:101) C ∗ ( X BN ( L, s )) are exactly thecanonical cycles α ( L, o ) ∈ C BN ( L, s ) under the identification of Theorem 1.
Proposition 3.23.
By a finite sequence of handle cancellations, the framedflow category C BN ( L, s ) can be transformed into the discrete category consistingof objects { x α ( L, o ) } o . The following proof is a flow category level analogue of the argument givenby Wehrli in [Weh08, Remark 5.4]. A more detailed argument can be found in[Lew09], [Tur20] and [San20].
Proof.
Consider the underlying 4-valent graph G of L . An edge labeling of G isan assignment of either X or Y to each edge of G . We say an edge labeling is admissible if each crossing admits a resolution such that the two arc segmentscan be labeled accordingly (see Figure 8). For each XY -labeled resolution con-figuration x = ( D L ( u ) , x ), there is a unique admissible edge labeling such thatthe labels on the circles agree with the edge labeling. Since the relation givenin Figure 7 is closed under each admissible edge labeling, the set of all ad-missible edge labelings { e } gives a decomposition of the framed flow category C = C BN ( L, s ) into a disjoint union of framed subcategories C = (cid:71) e C e . Admissible edge labelings of G are separated into two types: (I) there is acrossing of G such that the four incident edges are labeled the same, and (II)every crossing of G have two incident edges labeled X and the other two labeled Y . The statement follows from the following three claims:18 laim 1. For each type (I) edge labeling e , the associated subcategory C e canbe transformed into an empty category by a sequence of handle cancellations. Take one crossing v that has four incident edges labeled the same. All objectsof C e can be paired as x (cid:31) y , so that x is obtained from y by surgering thearc that passes v . We have M ( x , y ) = { pt } by definition. Take one such pair x (cid:31) y . From Proposition 2.21, the pair ( x , y ) can be removed from C e with theeffect of modifying other relevant moduli spaces. For any other pair x (cid:48) (cid:31) y (cid:48) ,the effect on M ( x (cid:48) , y (cid:48) ) is the attaching of the product M ( x , y (cid:48) ) × M ( x (cid:48) , y ) . For the two multiplicands to be non-empty, we need x (cid:31) y (cid:48) and x (cid:48) (cid:31) y , whichimplies | x | > | y (cid:48) | = | x (cid:48) | − , | x (cid:48) | > | y | = | x | − . This forces | x | = | x (cid:48) | and | y | = | y (cid:48) | , so we have y ≺ x (cid:31) y (cid:48) . The crossing v is 0-resolved for y , y (cid:48) and 1-resolved for x , so the above relationimplies that the underlying resolution configurations of y and y (cid:48) are the same.However, labelings on the circles are determined by the edge labeling e , so wemust have y = y (cid:48) which is a contradiction. Hence the above product is empty,and M ( x (cid:48) , y (cid:48) ) remains unchanged. Thus all pairs can be cancelled and we willbe left with an empty category. Claim 2.
Type (II) edge labelings correspond bijectively to the orientations of L . This can be seen by reversing the process described in Algorithm 2.8.
Claim 3.
For each type (II) edge labeling e that corresponds to an orientation o , the associated subcategory C e consists of a single object x α ( L, o ) . Every crossing of G admits a unique admissible resolution with respect to e ,and this is exactly the orientation preserving resolution for the correspondingorientation o . There is only one object whose labeling agrees with e , which isexactly x α ( L, o ). Thus the proof is complete.Since the grading of the canonical objects are given by Proposition 2.11, weobtain:
Theorem 2. X BN ( L, s ) is move equivalent to the wedge of spheres X BN ( L, s ) (cid:39) (cid:95) o σ α ( L, o ) where o runs over all orientations on L . In particular, the stable homotopy typeof X BN ( L, s ) is independent of the total ordering of the crossings of L and thesign assignment s . Moreover it is a link invariant.
19y applying (cid:101) H ∗ on both sides, we immediately recover the structure theoremfor Bar-Natan homology (Proposition 2.10). Also we obtain properties whosecounterparts in Khovanov homotopy type are known to be true [LLS20, Theorem1, 11.]: Corollary 1.1.
For link diagrams
L, L (cid:48) , there are stable homotopy equivalences:1. X BN ( L (cid:116) L (cid:48) ) (cid:39) X BN ( L ) ∧ X BN ( L (cid:48) ) .2. X BN ( m ( L )) (cid:39) X BN ( L ) ∨ .Here m ( · ) denotes the mirror, and ( · ) ∨ denotes the Spanier-Whitehead dual.Proof. 1. Let
X, X (cid:48) , X (cid:48)(cid:48) be realizations of the Bar-Natan flow categories of
L, L (cid:48) and L (cid:116) L (cid:48) , with integers (cid:96), (cid:96) (cid:48) > − (cid:96) X (cid:39) X BN ( L ), Σ − (cid:96) (cid:48) X (cid:48) (cid:39)X BN ( L (cid:48) ) and Σ − (cid:96) − (cid:96) (cid:48) X (cid:48)(cid:48) (cid:39) X BN ( L (cid:116) L (cid:48) ). This is possible because (cid:96), (cid:96) (cid:48) can betaken arbitrarily large. From Theorem 2 it suffices to prove σ α ( L (cid:116) L (cid:48) , o (cid:116) o (cid:48) ) ≈ σ α ( L, o ) ∧ σ α ( L (cid:48) , o (cid:48) )for arbitrary orientations o, o (cid:48) of L, L (cid:48) respectively. Put σ α = σ α ( L, o ) , σ (cid:48) α = σ α ( L (cid:48) , o (cid:48) ) and (cid:101) σ α = σ α ( L (cid:116) L (cid:48) , o (cid:116) o (cid:48) ). From Proposition 2.11, we havegr h ( σ α ) + gr h ( σ (cid:48) α ) = gr h ( (cid:101) σ α )and so dim( σ α ) + dim( σ (cid:48) α ) = (gr h ( σ α ) + (cid:96) ) + (gr h ( σ (cid:48) α ) + (cid:96) (cid:48) ) = dim( (cid:101) σ α ) . Thus we have σ α ∧ σ (cid:48) α ≈ (cid:101) σ α . From Proposition 2.11, we havegr h ( σ α ( m ( L ) , m ( o ))) = − gr h ( σ α ( L, o )) . Thus the statement follows from Theorem 2 and the standard S -duality map S p ∧ S − p → S . Next, in order to endow C BN ( L, s ) the quantum grading , we aim to recover thelabels { , X } from { X, Y } on the framed flow category level. First let us roughlyconsider how this should proceed. Since the attaching maps run opposite to thedifferential d of C BN , the label set corresponding to a basis S ⊂ A should beregarded as its dual basis. Let { X ∗ , Y ∗ } be the dual basis of { X, Y = X − } ,and { (cid:48) , X (cid:48) } be the dual basis of { , X } . From( X, Y ) = ( X, (cid:18) − (cid:19) ,
20e have ( X (cid:48) , (cid:48) ) = ( X ∗ , Y ∗ ) (cid:18) − (cid:19) . So the basis change from { X ∗ , Y ∗ } to { X (cid:48) , (cid:48) } is realized by { X ∗ , Y ∗ } add Y ∗ to X ∗ −−−−−−−−−→{ X ∗ + Y ∗ , Y ∗ } negate Y ∗ −−−−−−→{ X ∗ + Y ∗ , − Y ∗ } = { X (cid:48) , (cid:48) } . On the framed flow category, adding Y ∗ to X ∗ should be realized by sliding anobject labeled X over Y . We denote this process by X Y. (+)
The basis change for ( A ⊗ k ) ∗ ∼ = ( A ∗ ) ⊗ k can be performed by repeating thisprocess for each tensor factor. On the framed flow category, this should berealized by a k -times batch of parallel handle slides. The following diagramdescribes this process when k = 2. X ⊗ Y Y ⊗ Y X (cid:48) ⊗ Y Y ⊗ Y X (cid:48) ⊗ Y Y ⊗ YX ⊗ X Y ⊗ X X (cid:48) ⊗ X Y ⊗ X X (cid:48) ⊗ X (cid:48) Y ⊗ X (cid:48) (+)(+) (+) (+) Negating Y ∗ should be realized by reversing the orientations of the correspond-ing cells. The above consideration is formalized as follows.Let D be an index n resolution configuration. Take any sign assignment s and consider the (unnormalized) framed flow category ¯ C BN ( D, s ). For each u ∈{ , } n , put k ( u ) = Z ( D ( u )) and fix a total ordering of Z ( D ( u )) as { Z i } k ( u ) i =1 .Any XY -labeling x of D ( u ) can be identified with a vertex v ∈ { , } k ( u ) underthe correspondence v i = 0 ⇔ x ( Z i ) = X. Then an object x = ( D ( u ) , x ) is identified with a pair of vertices ( u, v ), andthe object set of ¯ C BN ( D ) is identified with the union of the vertices of the 2 n subcubes each placed at a vertex of the n -cube.Take one u ∈ { , } n and focus on the k ( u )-subcube placed at u . Let { e i } be the standard basis of R k ( u ) . For each 1 ≤ i ≤ k ( u ), perform 2 k ( u ) − positivehandle slides parallel to e i , so that each x = ( u, v ) with v i = 0 is slid over x (cid:48) = ( u, v + e i ). In total we perform k ( u ) · k ( u ) − handle slides. We call suchprocedure the cubic handle slide at u .Now perform cubic handle slides on all of the subcubes (in any order). Let¯ C (cid:48) BN ( D, s ) denote the resulting framed flow category. We temporally regard anobject x (cid:48) of ¯ C (cid:48) BN ( D ) placed at ( u, v ) as an X (cid:48) Y -labeled resolution configuration( D ( u ) , x (cid:48) ) under the correspondence v i = 0 ⇔ x (cid:48) ( Z i ) = X (cid:48) . We formally express x (cid:48) = x (cid:48) ⊗ · · · ⊗ x (cid:48) k ( u ) where x (cid:48) i = x (cid:48) ( Z i ).21 YX X (cid:48) X ⊗ Y Y ⊗ Y X (cid:48) ⊗ Y Y ⊗ YX ⊗ X Y ⊗ X X (cid:48) ⊗ X (cid:48) Y ⊗ X (cid:48)− − (+) − + −− (+)(+)(+) (+) Figure 9: Handle slides for a basic merge configuration X ⊗ Y Y ⊗ Y X (cid:48) ⊗ Y Y ⊗ YX ⊗ X Y ⊗ X X (cid:48) ⊗ X (cid:48) Y ⊗ X (cid:48) Y YX X (cid:48) (+)(+)(+) (+) + − (+) Figure 10: Handle slides for a basic split configuration
Lemma 4.1.
Suppose D is a basic index resolution configuration. The resultof the performing cubic handle slides on ¯ C BN ( D, s ) is depicted as in Figure 9if D is a merge configuration, or as in Figure 10 if D is a split configuration.Here, the vertical arrows represent to the -dimensional moduli spaces (with thedescending direction, opposite to the direction of surgeries), and the attachedsigns represent the framings.Proof. Straightforward from the formula of Proposition 2.22.
Lemma 4.2.
Suppose D is an index resolution configuration (not necessarilybasic). The signed -dimensional moduli spaces of ¯ C (cid:48) BN ( D, s ) are given by Y ⊗ Y Y Y X (cid:48) ⊗ YY ⊗ X (cid:48) X (cid:48) Y ⊗ X (cid:48) X (cid:48) ⊗ Y Y ⊗ YX (cid:48) ⊗ X (cid:48) X (cid:48) X (cid:48) ⊗ X (cid:48)−−− (4.1) depending on whether D is a merge configuration or a split configuration. Here (cid:48) · x · y · · + − + − + − Figure 11: Cancelling pairs of 0-dimensional moduli spaces we have omitted the labels on the circles that are not involved in the surgery,where they are assumed to be identical before and after the surgery. Also, wehave omitted pairs of arrows that are signed oppositely.Proof.
We assume that D is a merge configuration, and that the cubic handleslides are performed at u = 0 first. The proof for the other cases proceedsimilarly. Put D = D (cid:116) D where D is basic index 1 merge configuration and D consists of disjoint k circles. Any object of ¯ C BN ( D ) can be identified witha triple ( u, v, w ) where ( u, v ) ∈ { } × { , } ∪ { } × { , } and w ∈ { , } k . Wesay an arrow in ¯ C BN ( D ) is red (resp. blue ) if it corresponds to a merge of twocircles labeled X (resp. Y ). Before the handle slides, there are 2 k positive redarrows (1 , , w ) + −→ (0 , (0 , , w )and 2 k negative blue arrows(1 , , w ) − −→ (0 , (1 , , w ) . For each w , the arrows between objects (1 , ∗ , w ) and (0 , ( ∗ , ∗ ) , w ) are modifiedby the handle slides as in Figure 9, and the unpaired arrows are given by theleft side of (4.1). We must also consider arrows between objects (1 , ∗ , w ) and(0 , ( ∗ , ∗ ) , w (cid:48) ) where w < w (cid:48) . We claim that these arrows pairs into ones withopposite signs.Let x = (1 , v, w ) , y = (0 , v (cid:48) , w (cid:48) ) where v ∈ { , } , v (cid:48) ∈ { , } and w < k w (cid:48) .First consider the case when v = 0 and v (cid:48) < (1 , u = 0, the positive red arrow x = (1 , , w ) + −→ (0 , (0 , , w ) is copied to M ( x , y )as x ( − | v (cid:48)| + | w | −−−−−−−−→ y | v (cid:48) | + | w | times. Let x i = (1 , , w i ) ( w < w i < k − w (cid:48) , ≤ i ≤ k ) be the objects that x slides over, by the cubic handle slide at u = 1.From the above observation, M ( x , y ) also consists of a single arrow x − | v (cid:48)| + | w |− −−−−−−−−−→ y and is copied to M ( x , y ) by x (cid:57)(cid:57)(cid:75) x . At this point, M ( x , y ) consists of apair of two oppositely signed arrows. Figure 11 depicts this situation when v (cid:48) = (0 ,
0) and k = 2. By the same argument, M ( x , y ) consists of a pair of twooppositely signed arrows, and is copied to M ( x , y ) by x (cid:57)(cid:57)(cid:75) x . Continuinginductively, we see that the claim holds for this case.Next, consider the case v = 1 and v (cid:48) = (1 , v = 0 and v (cid:48) = (1 , X (cid:48) BN ( D, s ) of ¯ C (cid:48) BN ( D, s ). We reorientthe cells of ¯ X (cid:48) BN ( D, s ) by reversing the orientations of the cells σ x that corre-spond to objects x = ( u, v ) with | v | odd. We regard an object x of ¯ C (cid:48) BN ( D )placed at ( u, v ) as an 1 X -labeled resolution configuration ( D ( u ) , x ) under thecorrespondence v i = 0 ⇔ x ( Z i ) = X. Proposition 4.3. ¯ C (cid:48) BN ( D, s ) refines ¯ C ∗ BN ( D, s ) .Proof. From Lemma 4.2, the differential d of the associated cochain complex of¯ C (cid:48) BN ( D, s ) is given by the signed sum of the correspondences:1 ⊗ X ⊗ ⊗ X X ⊗ XX ⊗ ⊗ X ⊗ X X X ⊗ X − This is exactly the operations of the defining Frobenius algebra A , (compare(2.1) with ( h, t ) = (1 , C (cid:48) BN ( D, s ) is constructed by fixing a total ordering of the verticesof the n -cube and also one for each of the k ( u )-subcubes to determine the orderof the handle slides. A different set of choice yields a move equivalence betweenthe two framed flow categories, so in particular the resulting spectra are moveequivalent. Definition 4.4 ((1X-based) Bar-Natan spectrum) . Let L be a link diagramwith a fixed total ordering of its crossings. The framed flow category C (cid:48) BN ( L, s ) isconstructed similarly by performing handle slides on C BN ( L, s ). The associatedspectrum X (cid:48) BN ( L, s ) of C (cid:48) BN ( L, s ) is called the (1X-based) Bar-Natan spectrum of L . 24 roposition 4.5. There is an isomorphism (cid:101) C ∗ ( X (cid:48) BN ( L, s )) ∼ = −−→ C ∗ BN ( L, s ) that maps the dual cells of X (cid:48) BN ( L, s ) to the X -enhanced states of C ∗ BN ( L, s ) . Proposition 4.6.
Let f : X BN ( L, s ) (cid:39) −→ X (cid:48) BN ( L, s ) , be the composition of all of the move equivalences corresponding to the abovedescribed handle slides. Then the following diagram commutes: (cid:101) C ∗ ( X BN ( L, s )) (cid:101) C ∗ ( X (cid:48) BN ( L, s )) C ∗ BN ( L, s ) ∼ = f ∗ ∼ = Here left diagonal arrow is the isomorphism given in Theorem 1, and the rightdiagonal arrow is the one given in Proposition 4.5.Proof.
Consider the dual bases { X ∗ , Y ∗ } of { X, Y } and { X (cid:48) , (cid:48) } of { X, } . Fromthe construction of f , we see that the following diagram commutes. (cid:101) C ∗ ( X BN ( L, s )) (cid:101) C ∗ ( X (cid:48) BN ( L, s )) (cid:76) u Z { X ∗ , Y ∗ } ⊗ k ( u ) (cid:76) u Z { X (cid:48) , (cid:48) } ⊗ k ( u ) f ∗ ∼ = ∼ = The result follows by duality.
Definition 4.7.
Let L be a link diagram with n ± positive / negative crossings.The quantum grading function gr q : Ob( C (cid:48) BN ( L )) → Z is defined as gr q ( x ) := gr h ( x ) + deg( x ) + n + − n − for x = ( D L ( u ) , x ).This definition is identical to Definition 2.6, so the isomorphism of Propo-sition 4.5 preserves both homological and quantum gradings. It is natural toexpect that the filtration on C BN ( L ) also lifts to X (cid:48) BN ( L ). Recall that the (de-scending) filtration on C BN is well-defined by the fact that the differential d a) The ladybug configuration (b) The Hopf link configuration is quantum grading non-decreasing. The corresponding condition on X (cid:48) BN ( L )should be that each cell does not attach to cells of higher quantum grading. Thisis not true in general, which we have seen in Figure 9 or in Figure 10, wherethe diagonal arrows increase the quantum grading. However, for 0-dimensionalmoduli spaces, Lemma 4.2 tells us that such moduli spaces consist of pairsof points (arrows) with opposite signs. Each such pair can be eliminated byperforming the Whitney trick on C (cid:48) BN ( L ) (Proposition 2.23). We conjecturethat the same applies to higher dimensional moduli spaces by performing the extended Whitney trick (Remark 2.24). Conjecture 1.
The framed flow category C (cid:48) BN ( L, s ) can be modified (possiblyafter altering the framing) so that there are no moduli spaces that increase thequantum grading. This gives an ascending filtration on C = C (cid:48) BN ( L, s ) as: ∅ = F m C ⊂ · · · ⊂ F j C ⊂ F j +2 C ⊂ · · · ⊂ F M C = C and induces a filtration on the associated spectra X = X (cid:48) BN ( L, s ) as: { pt } = F m X ⊂ · · · ⊂ F j X ⊂ F j +2 X ⊂ · · · ⊂ F M X = X . The filtration { F j X } on X and the (descending) filtration { F j C } on C = C BN ( L, s ) correspond as (cid:101) C ∗ ( F j X ) = C/F j +2 C, (cid:101) C ∗ ( X /F j X ) = F j +2 C. We also expect that, if Conjecture 1 is true, Corollary 1.1 can be refined sothat the stable homotopy equivalences respect the filtration. Also, for a linkcobordism S , there should be a filtered morphism between the correspondingfiltered spectra. Recall that the Khovanov complex and the Bar-Natan complexare related by F j C BN /F j − C BN = C ∗ ,j Kh for each j ∈ Z + | L | . Our secondconjecture is that this also lifts to the spatial level: Conjecture 2.
For each j ∈ Z + | L | , the quotient F j X /F j − X is stably homo-topy equivalent to the j -th wedge summand X j Kh ( L ) of the Khovanov spectrum X Kh ( L ) . The two conjectures are true when n = 0 ,
1. Examples for n = 2 follows. Example 4.8 (The ladybug configuration) . The ladybug configuration (Fig-ure 12a) is exceptional in the construction of Khovanov homotopy type, forit is the only index 2 basic resolution configuration whose corresponding flow26ategory contains I (cid:116) I in its 1-dimensional moduli space. Figure 13 depictsthe framed flow category C (cid:48) BN ( D ) for the ladybug configuration D . Verticalarrows correspond to the 0-dimensional moduli spaces of C (cid:48) BN ( D ), and thethickened ones are the ones that existed in C BN ( D ). The attached signs corre-spond to the signs of 0-dimensional moduli spaces (not the signs of the differen-tial). In C BN ( D ) there are two 1-dimensional moduli spaces M ( X (11) , X (00) ) = M ( Y (11) , Y (00) ) = I , which are depicted in the figure as dashed intervals XX (10) XX (01) and 11 (10) (01) . The handle slides in the top and bottom levels copy the twointervals, and we get M ( X (11) , (00) ) = I (cid:116) I . Moreover, the handle slidesin the intermediate level produce four more 1-dimensional moduli spaces in M ( X (11) , (00) ), that are XX (10) X (10) , 1 X (10) (10) , XX (01) X (01) , 1 X (01) (01) .There are four oppositely signed pairs of vertical arrows. By applying Whitneytricks to these arrows, the six intervals in M ( X (11) , (00) ) concatenate togetherto form two disjoint intervals. We have gr q ( X (11) ) = gr q (1 (00) ) = 1 and theboundary objects X (10) , X (01) , X (10) , X (01) also lies in gr q = 1. There areno other quantum grading preserving 1-dimensional moduli spaces. Thus by cut-ting off quantum grading decreasing moduli spaces, we recover the Khovanovflow category for the ladybug configuration. Example 4.9 (The Hopf link configuration) . Let D be the resolution con-figuration given in Figure 12b. Figure 14 depicts the framed flow category C (cid:48) BN ( D ). Let us focus on the 1-dimensional moduli space M ( XX (11) , (00) )where gr q ( XX (11) ) = 0 < gr q (11 (11) ) = 2. After performing the Whitney trickson the four pairs of oppositely signed arrows, we see that the four intervals of M ( XX (11) , (00) ) concatenate to form a single circle C . Is C trivially framedso that it can be removed by the extended Whitney trick?To answer this question, one must specify how the 1-dimensional modulispaces of C BN are framed. In [LS14c], a specific framing of the cube flow categoryis given, where the 1-dimensional moduli spaces are framed using the framepaths . Suppose we adopt this framing in the construction of C BN . A framing ofa 1-dimensional moduli space M is encoded into a function f r : π ( M ) → π ( SO ( d )) = Z / d ≥
3. From the formulasgiven in [JLS17, Proposition 3.3, 3.4] and [LOS18, Proposition 3.8], one cancompute f r ( C ) = 0, meaning that C has the non-trivial framing (see [JLS17,Remark 3.1]). If we instead adopt the non-standard frame path for the sub-category corresponding to the XY -triad ( D ; Y, Y ), then we get f r ( C ) = 1 andhence C can be removed.Example 4.9 suggests that, to attack Conjecture 1, one must first specify anappropriate framing for the high dimensional moduli spaces. Question 4.10.
An alternative construction of Khovanov homotopy type isgiven in [LLS20], where the original construction is reinterpreted as a functorfrom the cube to the Burnside category. Does Bar-Natan homotopy type alsoadmit such alternative construction? 27
010 0111 − X X X XX X XX XX − − −−−− − Figure 13: C (cid:48) BN for the ladybug configuration28
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