Almost-extreme Khovanov spectra
Abstract
We introduce a functor from the cube to the Burnside 2-category and prove that it is equivalent to the Khovanov spectrum given by Lipshitz and Sarkar in the almost extreme quantum grading. We provide a decomposition of this functor into simplicial complexes. This decomposition allows us to compute the homotopy type of the almost extreme Khovanov spectra of diagrams without alternating pairs.
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Selecta Mathematica (2021) 27:95 https://doi.org/10.1007/s00029-021-00706-6 Selecta Mathematica New Series Almost-extreme Khovanov spectra Federico Cantero Morán1·Marithania Silvero2 Accepted: 10 August 2021 / Published online: 29 September 2021 © The Author(s) 2021 Abstract We introduce a functor from the cube to the Burnside 2-category and prove that it is equivalent to the Khovanov spectrum given by Lipshitz and Sarkar in the almostextreme quantum grading. We provide a decomposition of this functor into simplicial complexes. Thisdecompositionallowsus tocompute thehomotopytype ofthe almostextreme Khovanov spectra of diagrams without alternating pairs. Mathematics Subject Classification Primary 57M25 ·55P42 1 Introduction Khovanov homology is a powerful link invariant introduced by Mikhail Khovanov in [4] as a categorification of the Jones polynomial. More precisely, given an oriented diagram Drepresenting a link L, he constructed a finite Z-graded family of chain complexes ··· Ci,j(D)diCi+1,j(D)di+1Ci+2,j(D)··· whosebigradedhomologygroups, Khi,j(D),arelinkinvariants.Thegroups Khi,j(L) are known as Khovanov homology groups of L, and the indexes iand jas homological and quantum gradings, respectively. Both authors were supported by the Spanish projects MTM2016-76453-C2, MDM-2014-0445 and FEDER. The first author was partially supported by project PID2019-108936GB-C21. The second author was partially supported by projects PID2020-117971GB-C21, US-1263032, and IJC2019-040519-I/AE. BFederico Cantero Morán [email protected] Marithania Silvero [email protected] 1Departamento de Matemáticas, Universidad Autónoma de Madrid and ICMAT, Madrid, Spain 2Departamento de Álgebra, Universidad de Sevilla and IMUS, Sevilla, Spain
95 Page 2 of 45 F. C. Morán, M. Silvero In [7] Lipshitz and Sarkar refined this invariant to obtain, for each quantum grading j, a spectrum Xj Dwhose stable homotopy type is a link invariant and whose cohomology is isomorphic to the Khovanov homology of the link, i.e., H∗(Xj D)∼ =Kh∗,j(L). Together with Lawson [6] they gave a neat construction of this spectrum as the realization of a 2-functor Fjfrom the cube category to the Burnside 2-category (see also [5,8]). Shortly after, another geometric realization of the (minimal) extreme Khovanov homology was given [3] in terms of an independence simplicial complex constructed from the link diagram. This construction was later shown to coincide with the one of Lipshitz and Sarkar [1] as follows: functors from the cube to the Burnside 2- category can be understood as generalizations of simplicial complexes; such a functor is a simplicial complex if it factors through the category of sets and takes values on singletons and the empty set. Onthe otherhand, theKhovanovspectrum ofa linkinits maximalextremequantum grading is Spanier-Whitehead dual to that of its mirror image in minimal grading. This construction was exploited in [10] to make explicit computations of the Khovanov spectrum in (maximal) extreme quantum grading. In 2018 the second author, together with Przytycki applied these techniques one step further, to the almost-extreme quantum grading. They restricted to the study of 1-adequate link diagrams, i.e., those whose 1-resolution contains no chords with both endpoints in the same circle (the extremal homology of these diagrams has been computed in [2,9,12]), and built a pointed semi-simplicial set whose homology is isomorphic to the almost-extreme Khovanov homology of the diagram: Theorem ([11]) If D is a 1-adequate link diagram, then (i) The functor F jalmax Dgives rise to a pointed semi-simplicial set. (ii) The realization of F jalmax Dhas the homotopy type of a wedge of spheres and possibly a copy of a (de)suspension of RP2. Their construction relates to the functor of Lawson, Lipshitz and Sarkar as follows: the category of pointed sets includes into the Burnside 2-category, and if a functor from the cube to the Burnside 2-category factors through it, then it gives rise to a pointed (augmented) semi-simpicial set. Simplicial complexes and pointed semi-simplicial sets are simpler objects than functors to the Burnside 2-category, to which many classical tools can be applied, so these results prompt the question Are there simpler models of the spectra of Lipshitz and Sarkar in the almostextreme quantum grading? Unfortunately, it turns out that Fjalmax Dgives rise to a pointed semi-simplicial set if and only if Dis 1-adequate. However, in this paper we overcome this problem by introducing a new functor MD, whose realization coincides with that of Fjalmax D, and gives rise to an (augmented) pointed semi-simplicial set in a broader number of cases. This is achieved by constructing a natural transformation between MDand Fjalmax Dand proving the following result:
Almost-extreme Khovanov spectra Page 3 of 45 95 Theorem 1 The natural transformation is a homology isomorphism, and therefore the realizations of MDand F jalmax Dare homotopy equivalent. The functor MDcan be understood as a “change of basis” of Fjalmax D: The relevant generators of the Khovanov chain complex in its almost-extreme degree (which is the same as the chain complex of Fjalmax D) are indexed by the circles of the state resolutions of D, whereas the generators of the chain complex associated to MDare indexed by the edges and connected components of certain graph whose vertices are the circles of the state resolutions. A crucial property at the almost-extreme quantum grading is that the relevant involved graphs are forests, and therefore both bases have the same size (as it should be). The functor MDis simpler than Fj Din the sense that it takes values in morphisms sendinggenerators to(multiples of)generators,whereas Fj Dtakesvaluesin morphisms sending generators to sums of possibly different generators. This simplified new basis allows us to decompose MDin terms of independence simplicial complexes of simpler link diagrams (Proposition 7.5 and Remark 7.6). The last part of the paper is devoted to make explicit computations using this decomposition. One of the advantages of MDover Fjalmax Dis that it gives rise to an augmented pointed semi-simplicial set if and only if the 1-resolution of the diagram contains no alternating pairs (i.e., two chords whose endpoints alternate along the same circle). Moreover, the forementioned decomposition allows us to compute the homotopy type of the realization of MDin these cases: Theorem 2 Let D be a diagram whose 1-resolution contains no alternating pairs. (i) The functor MDgives rise to an augmented pointed semi-simplicial set. (ii) If D is not 1-adequate then the realization of MDis homotopy equivalent to a wedge of spheres. In[10] the secondauthor conjectured(and provedfor severalcases) thatthe extreme Khovanov spectrum is always a wedge of spheres. If this conjecture were true, then the above decomposition would give an upper bound for the cone length of the spectrum Xjalmax D(Remark 7.8). This bound seems to be far from optimal, as the computations of this paper and [11] produce spectra of cone length at most 2, so we rise the following question: Are there diagrams whose almost-extreme Khovanov spectrum is not homotopy equivalent to a wedge of spheres and possibly (de)suspensions of projective planes? The structure of the paper is as follows. In Sections 2 and 3 we recall the categorical notions that will be used along the paper. In Section 4 we develop some results on link diagrams which allow, in Section 5, to introduce the functor MDand prove Theorem 1. In Section 6 we prove the first part of Theorem 2. In Section 7 we decompose MDin terms of independence simplicial complexes and present three skein sequences which allow us to determine, in Section 8, the homotopy type of MDfor all links with no alternating pairs, proving the second part of Theorem 2.
95 Page 4 of 45 F. C. Morán, M. Silvero 2 The Burnside 2-category A 2-category is a higher category that generalizes the notion of category, in the sense that on top of the objects and morphisms there are also 2-morphisms. More precisely, a 2-category consists of the data of (1) a collection of objects, (2) for each pair of objects X,Y, a category of morphisms hom(X,Y), (3) for each object X, an object IdXof hom(X,X)and (4) for each triple of objects X,Y,Z, a composition functor hom(Y,Z)×hom(X,Y)→hom(X,Z) which is associative and unital. The objects in hom(X,Y)are called 1-morphisms and the morphisms in hom(X,Y)are called 2-morphisms. Recall that given two finite sets X,Y,aspan from Xto Yis a triple (Q,s,t)where Qis a finite set, and sand tare a pair of functions Xs ←− Qt −→ Y. We say that a span is free (resp. very free) if the source map sis injective (resp. bijective). Given two spans Xs ←− Qt −→ Yand Xs ←− Qt −→ Y,afibrewise bijection between them is a bijection τ:Q→Qsuch that s◦τ=sand t◦τ=t: Q s t τ XY Q st The composition of two fibrewise bijections of spans is the composition of bijections, and the identity morphism of a locally finite span is the identity bijection. This defines a category of spans from the set Xto the set Y, which we note by B(X,Y). Definition 2.1 The Burnside 2-category, denoted by B, is the 2-category whose objects are finite sets, and the category of morphisms from a set XtoasetYis given by B(X,Y), the category of spans from Xto Y; in other words, the 1-morphisms are given by spans and 2-morphisms are fibrewise bijections. Let X1,X2and X3be three sets in B. Given a span (Q1,s1,t1)in B(X1,X2)and a span (Q2,s2,t2)in B(X2,X3), their composition is given by their fibre product (Q1×X2Q2,s,t): Q1×X2Q2 s t Q1 s1t1 Q2 s2t2 X1X2X3.
Almost-extreme Khovanov spectra Page 5 of 45 95 The composition of two fibrewise bijections τ1:Q1→Q 1and τ2:Q2→Q 2is also given by their fibre product τ1×τ2:Q1×X2Q2→Q 1×X2Q 2as follows Q1 τ1 Q2 τ2 X1X2X3 Q 1Q 2 Q1×X2Q2 τ1×τ2 XY, Q 1×X2Q 2, and the identity on a set Xis the span XId ←XId →X. Remark 2.2 Note that a span s=(Q,s,t)from Xto Yis determined by the matrix whose entries are the sets sx,y:= s−1(x)∩t−1(y), for every x∈X,y∈Y.Givinga fibrewise bijection τfrom a span sto a span sis the same as giving, for each x∈X and each y∈Y, a bijection τx,yfrom sx,yto s x,y. The composition of s:X→Yand s:Y→Zis the span s with s x,z=y∈Ysx,y×s y,z. The span sis determined by the formal sum s(x)= y∈Y sx,y·y, for each x∈X. If a coefficient sx,yis not specified, we understand that sx,y={x} (for example, if s(x)=y+z, then s(x)={x}·y+{x}·z). The category of finite sets Set can be mapped into the category of pointed finite sets Set•by sending Xto X+:= X∪{∗}, and a morphism f:X→Yto the morphism f+that coincides with fon Xand sends the basepoint of X+to the basepoint of Y+. Moreover, the category Set•sits inside Bby sending a pointed finite set Xto X{∗}, and a morphism f:X→Yto the span X{∗} ←Xf−1(∗)f →Y{∗}. The inclusions Set →Set•→Binduce equivalences between the following 2-subcategories of B: •The essential image of Set•in Bis the 2-subcategory of free spans. •The essential image of Set in Bis the 2-subcategory of very free spans. Fix now a commutative ring Rwith unit, and let R-Mod be the category of modules over R. There is a functor A:B→R-Mod (2.1) sending the finite set Xto the free R-module RX, and a span s:X→Yto the homomorphism f:RX→RYgiven, for every x∈X,by f(x)= y∈Y |sx,y|·y.
95 Page 6 of 45 F. C. Morán, M. Silvero Note that every pair of spans connected by a 2-morphism are sent to the same homomorphism, and therefore the functor Ais well-defined. 3 Cubes of pointed sets and augmented semi-simplicial pointed sets The n-dimensional cube 2nis the partially ordered set (poset) whose elements are n-tuples u=(u1,...,un)∈{0,1}nendowed with the standard partial order so that u≥vif ui≥vifor all i. Following [6], we regard 2nas the category whose objects are the elements of this poset and, for any two such elements u,v, the morphism set Hom(u,v)has a single element ϕu,v if u≥v, and is empty otherwise. This category has an initial element 1=(1,1,...,1)and a terminal element 0=(0,0,...,0). The poset map |·|: 2n→Ngiven by |u|=iuiassigns a grading to each vertex in the cube. We write uvif u>vand |u−v|=1, and we write uivif, additionally, uand vdiffer in the ith-coordinate, i.e., if ui=1, vi=0 and uj=vj for i= j. Given n∈Z,n≥−1, the finite ordinal [n]is the linearly ordered set {0< 1<...<n}.Theaugmented semi-simplicial category,inj∗, has as objects these finite ordinals, and as morphisms injective order-preserving maps between them. The inclusion ∂i:[n−1]→[n]that forgets the i-th element is called ith-face map. Definition 3.1 Given a category C,letCop denote its opposite category, obtained by reversing the morphisms. An n-dimensional cube of pointed (finite) sets is a functor F:2n→Set•, while an augmented semi-simplicial pointed (finite) set is a functor X:op inj∗→Set•. As usual, we write Xnand ∂ifor X([n])and X(∂i), respectively. These two categories are related by a functor λ:2n→op inj∗which maps every vertex of the cube uto the ordinal [|u|−1], and every morphism uivto the opposite of the j<iuith-face map. Taking left Kan extension along the functor λ:2n→op inj∗defines a functor :Set2n •−→ Setop inj∗ •,(3.1) whose value on a cube of pointed sets Fis defined explicitely as follows: its set of k-simplices is (F)k=|u|=k+1F(u), for every u∈2n, and the ith-face map ∂i:(F)k→(F)k−1is the union |u|=k+1F(u>u[i]), where u[i]is the vertex of the cube 2nobtained by replacing the (i+1)th one in uby a zero. Recall from [5, Definition 4.1] that a strictly unitary lax 2-functor F from the cube 2nto the Burnside category Bconsists of the data: (1) a finite set F(v) ∈Ob(B), for each v∈{0,1}n. (2) a finite span F(ϕu,v):F(u)→F(v) in B, for each morphism ϕu,v in 2n. (3) a 2-isomorphism Fu,v,w from F(ϕv,w)◦F(ϕu,v)to F(ϕu,w), for each decomposition ϕu,w =ϕv,w ◦ϕu,v;
Almost-extreme Khovanov spectra Page 7 of 45 95 satisfying that, for every u>v>w>z, the following diagram commutes: F(ϕw,z)×F(w) F(ϕv,w)×F(v) F(ϕu,v)Id ×Fu,v,w Fv,w,z×Id F(ϕw,z)×F(w) F(ϕu,w) Fu,w,z F(ϕv,z)×F(v) F(ϕu,v)Fu,v,zF(ϕu,z). In practice, when defining a strictly unitary lax 2-functor, we use the following result, from [5, Section 4]. We keep the notation u•vfor the morphism uv: Lemma 3.2 A strictly unitary lax 2-functor from the cube category 2nto the Burnside category Bis uniquely determined (up to natural isomorphism) by the following data: (D1) for each vertex v∈{0,1}n, a finite set F(v) ∈Ob(B); (D2) for each u v, a finite span F(ϕu,v):F(u)→F(v) in B; (D3) for each two-dimensional face v• u• •w v•of the cube, a 2-morphism Fu,v,v,w :F(ϕv,w)×F(v) F(ϕu,v)→F(ϕv,w)×F(v)F(ϕu,v), satisfying the following two conditions: (C1) for every two-dimensional face v• u• •w v•,F u,v,v,w =F−1 u,v,v,w, (C2) for every three-dimensional face v• • w • u • • • v • • w•z v • • w •, the following commutes: F(ϕw,z)×F(w )F(ϕv,w )×F(v) F(ϕu,v ) Fv,w,w,z×Id Id×Fu,v,v,w F(ϕw,z)×F(w)F(ϕv,w)×F(v) F(ϕu,v ) Id×Fu,v,v,w F(ϕw,z)×F(w )F(ϕv,w )×F(v)F(ϕu,v) Fv,w ,w,z×Id F(ϕw,z)×F(w)F(ϕv ,w)×F(v )F(ϕu,v ) Fv,w,w,z×Id F(ϕw,z)×F(w) F(ϕv,w )×F(v)F(ϕu,v) Id×Fu,v,v,w F(ϕw,z)×F(w) F(ϕv,w)×F(v )F(ϕu,v ). Definition 3.3 [5]Anatural transformation α:F→Gbetween two strictly unitary lax 2-functors F,G:2n→Bis a strictly unitary lax 2-functor H:2n+1→Bsuch that H|2n×{1}=Fand H|2n×{0}=G. For every u∈2n, we denote by αuthe morphism H(ϕ(u,1),(u,0)). For every uv∈2n, we denote αu,v the 2-morphism H(u,1),(u,0),(v,1),(v,0). We write CDfor the category whose objects are strictly unitary lax 2-functors from Dto Cand whose morphisms are natural transformations between them. When C=B and D=2nthe objects are called Burnside cubes.
95 Page 8 of 45 F. C. Morán, M. Silvero 3.1 Realizations and totalizations Let Cbe a model category, as the category of (pointed) topological spaces Top (resp. Top•), the category of spectra Sp, or the category Ch(R)of chain complexes of R- modules, with Ra commutative ring. The totalization of a functor F:2n→Cis the object of Cgiven by Tot F=hocofib hocolim F|2n{0}→F(0). This defines a functor Tot :C2n−→ C. On the other hand, the relative realization of a functor X:op inj∗→Cis the object in Cgiven by X=hocofib hocolim X|inj∗{[−1]}→X−1, which defines a functor −:Cop inj∗−→ C. A semi-simplicial object X:op inj →Cmay be seen as an augmented semi-simplicial set by defining X−1=∗, the final object of C. Its relative realization is the suspension of the realization of X: X∼ =|X|. Recall now the functor λ:2n→op inj∗mapping every u∈2nto [|u|−1], and every morphism uivto the opposite of j<iuith-face map. Since λis cofinal, λ(2n{0})=inj∗{[−1]} and λ(0)=[−1], we have that (F)≃Tot F.(3.2) 3.2 From Burnside cubes to chain complexes Let Rbe a ring, and let Ch(R)be the category of chain complexes of R-modules. There is a functor K:B−→ R-Mod −→ Ch(R), where the first functor is A, defined in (2.1), and the second functor sends an R-module to the chain complex given by the R-module concentrated at degree 0.
Almost-extreme Khovanov spectra Page 9 of 45 95 We define the functor C∗(−; R)as the composition C∗(−; R):B2n−→ Ch(R)2nTot −→ Ch(R), where the first functor is the result of composing a 2-functor F:2n→Bfrom B2n with K. Its value at the Burnside cube Fcan also be described as the chain complex whose k-cochains are Ck(F;R)= u∈2n,|u|=k RF(u) and whose differential is ∂= |u|=k,|v|=k−1 (−1)j<iuiA◦F(uiv). Observe that since B→R-Mod sends 2-morphisms to identities, a natural transformation between Burnside cubes induces a morphism of chain complexes. 3.3 From Burnside cubes to spectra In [6], Lawson, Lipshitz and Sarkar gave an explicit construction of the totalization functor for cubes in the Burnside category. They associated, to each cube F:2n→B, a spectrum Tot F, and to each map f:F→Gof Burnside cubes, amapTotf:Tot F→Tot G, both well-defined up to homotopy. Additionally, the homology H∗(Tot F;R)was isomorphic to the homology of C∗(F;R), and the map induced by Tot fon homology was the one induced by C∗(f;R). Remark 3.4 If C∗(f;R)is a homology isomorphism then Tot fis a homotopy equivalence of spectra. On the other hand, in [1] the authors showed that if a cube F:2n→Bfactors through some functor ˜ F:2n→Set•, then Tot F=∞Tot ˜ F, where the homotopy colimit is taken in pointed topological spaces. Therefore, by (3.2), we have: Proposition 3.5 If a Burnside cube F :2n→Bin the Burnside category factors through Set•, then the pointed semi-simplicial set ( ˜ F)satisfies Tot F≃∞( ˜ F). Remark 3.6 The spectrum Tot Fis denoted |F|in [6]. The chain complex C∗(F;R) is denoted Tot Fin [5]. We reserve the notation Tot for the totalization of a cube, ·for the relative realization of an augmented semi-simplicial object and |·|for the realization of a semi-simplicial object.
95 Page 16 of 45 F. C. Morán, M. Silvero –If(ej(v), ek(v)) is an alternating pair, then (ej(w), ek(w)) is an alternating pair. –If(ej(v), ek(v), el(v)) is an alternating triple, then either (ej(w), ek(w), el(w)) is an alternating triple, or (ei(w), ej(w), ek(w), el(w)) is a mixed alternating pair. –If(ej(v), ek(v), el(v), em(v)) is a mixed alternating pair, then (ej(w), ek(w), el(w), em(w)) is a mixed alternating pair. –If(ej(v), ek(v)) is an alternating pair and el(v) is a bichord, then either (ej(w), ek(w)) is an alternating pair and el(w) is a bichord or (ei(w), ej(w), ek(w), el(w)) is a mixed alternating pair, or (ei(w), el(w)) and (ej(w), ek(w)) are disjoint alternating pairs. Lemma 4.10 Let wv>u be three states with (w) =(v) −1. Then: (1) If D(v)ucontains a bichord, then D(w)ucontains two non-parallel bichords. (2) If D(v)ucontains an alternating pair, then D(w)ucontains an alternating triple or an alternating pair and a bichord. Proof Since (w) =(v) −1, wivis a merging for some bichord ei(w).The lemma follows from Remark 4.7 together with the following facts (i= j= k= l): •If ej(v) is a bichord, then ej(w) and ei(w) are non-parallel bichords. •If (ej(v), ek(v)) is an alternating pair, then either (ej(w), ek(w)) is an alternating pair and ei(w) is a bichord, or (ei(w), ej(w), ek(w)) is an alternating triple. Proof of the sufficient condition of Proposition 4.8 Fix a chain 1=u0i1u1i2...ikuk=u. Consider first the case (u)>0 and write mfor be the minimal number such that um−1imumis a merging. Therefore, eim(um−1)∈D(um−1)uis a bichord and applying Lemma 4.9 recursively, we deduce that D(1)ucontains a bichord or an alternating pair. Consider now the case (u)>1 and write m<pfor the first two indices such that um−1imumand up−1ipupare mergings. Therefore, (um)=(up−1), and applying recursively Lemma 4.9 to the bichord eip(up−1)∈D(up−1)uwe deduce that D(um)ucontains a bichord or an alternating pair. Apply once Lemma 4.10 and deduce that D(um−1)ucontains at least one of the following: two non-parallel bichords or an alternating triple or an alternating pair and a bichord. Finally,applyingineachcaserecursivelyLemma4.9wededucethat D(1)ucontains at least one of the following configurations: (1) Two non-parallel bichords. (2) An alternating pair and a bichord. (3) An alternating triple. (4) Two disjoint alternating pairs. (5) A mixed alternating pair.
Almost-extreme Khovanov spectra Page 17 of 45 95 Fig. 6 An example of D(u)for the case when (u)=1and |π0G(u)|=|π0G(1)|.The cycles c1and c2(of length 4 and 5, respectively) divide zinto four arcs. The ladybug set corresponds to the thickened arcs, i.e. L(u)={P,Q} 4.4 Ladybug set Let u∈2nso that (u)=1. If |π0G(u)|=|π0G(1)|−1, recall from Lemma 4.3(2a)that G(u)contains a unique cycle. We define the ladybug set of u,L(u),as the singleton consisting of the connected component of G(u)containing that cycle. Consider now the case when |π0G(u)|=|π0G(1)|. Then, by Lemma 4.3(2b), G(u)contains two cycles c1and c2sharing a common vertex z. Write ai,bifor the edges of ciadjacent to z,fori=1,2(ifciis a loop, then ai=bi). The chords a1,b1,a2,b2divide the circle zinto four arcs (see Fig. 6). We define the ladybug set L(u)of D(u)as the set whose elements are the two arcs of zthat are reached by traveling along any of the chords a1,b1,a2,b2until z, and then taking a right turn. Remark 4.11 In the case when |π0G(u)|=|π0G(1)|and ai=bifor i=1,2, the set L(u)coincides with the right pair giving rise to the celebrated ladybug matching defined in [7]. Lemma 4.12 If u >v,|π0G(v)|=|π0G(1)|and (u)=(v) =1, then each arc in L(u)intersects exactly one of the two arcs in L(v) and viceversa. Proof Let ={ei(u)|ui= vi}⊂ ¯ E(u). Since (u)=(v), all elements in are loops and no pair among them constitute an alternating pair. By Lemma 4.3(2b),G(u)contains two cycles c1and c2sharing a common vertex zu. Write zvfor the circle in D(v) with the same property. Then zvis obtained from zuby performing surgery along the loops in with their endpoints in zu. Write L(u)={P,Q}and let P(resp. Q) be the subset of loops of so that at least one of their endpoints lies in P(resp. Q). The disposition of c1and c2(stated in Lemma 4.3(2b)) implies that P∩Q=∅. Define the following subsets of R2: p= ei(u)∈P ei(u)and q= ei(u)∈Q ei(u). The endpoints of each loop ei(u)∈P(resp. ei(u)∈Q) separate zuinto two arcs: write difor the unique arc in zuwhich is disjoint from Q(resp. from P). Define
95 Page 18 of 45 F. C. Morán, M. Silvero the following subsets of R2: dP= ei(u)∈P diand dQ= ei(u)∈Q di. As there are no alternating pairs in , we deduce that dP∩dQ=∅ and (p∪dP)∩(q∪dQ)=∅ As a consequence, we can consider two disjoint open subsets A,Bof R2such that P∪p∪dP⊂AQ∪q∪dQ⊂B. Each of the arcs of L(v) is obtained by doing surgery on each of the arcs of L(u). The surgery performed on Pis supported in A, while the surgery perfomed on Q is supported in B. Since A∩B=∅, it is possible to define P(resp. Q)asthe arc of L(v) obtained from P(resp. Q). Thus, Pintersects P,Qintersects Qand P∩Q=Q∩P=∅, as desired. 4.5 The bijection u,v Given uivso that (u)=(v) =1, then |π0G(u)|=|π0G(v)|, and therefore it is possible to define a bijection ρu,v :L(u)→L(v) in the following way: (1) If |π0G(v)|=|π0G(1)|−1, then ρu,v maps the unique element in L(u)to the unique element in L(v). (2) If |π0G(v)|=|π0G(1)|, then by Lemma 4.12 each of the two arcs in L(u) intersects one of the two arcs in L(v). Write L(u)={Pu,Qu}and L(v) = {Pv,Qv}, labelling the arcs so that Pu∩Pv=∅and Qu∩Qv=∅as subsets of R2(see Fig. 7). The bijection ρu,v :L(u)→L(v) is given by sending Puto Pvand Quto Qv. In particular, in those cases when none of the endpoints of the chord ei(u)lie in Punor Qu,ρu,v becomes the identity. The following lemma follows immediately from Lemma 4.12: Lemma 4.13 Let u v,vwso that (u)=(w) =1. Then the following diagram commutes L(u)ρu,v ρu,v L(v) ρv,w L(v) ρv,w L(w).
Almost-extreme Khovanov spectra Page 19 of 45 95 (a) (b) Fig. 7 Two examples illustrating the bijection ρu,v for the case uiv,(u)=(v) =1and|π0G(v)|= |π0G(1)|, with the edge ei(u)depicted in light colour. In both cases ρu,v :L(u)→L(v) maps Puto Pv and Quto Qv Fig. 8 An exampleillustrating thebijection μi(u)for thecaseuiv,(u)=0,(v) =1and|π0G(v)|= |π0G(1)|. The edge ei(u)is depicted in light colour. The bijection μi(u):Oi(u)→L(v) sends z1and z2 to Pand Q, respectively 4.6 The bijection i(u) Given uivwith (u)=0, (v) =1 (i.e., a merging), we define a function μi(u):Oi(u)→L(v) as follows: (1) If |π0G(v)|=|π0G(1)|−1, then |L(v)|=1 and we define μi(u)as the unique constant function. (2) If |π0G(v)|=|π0G(1)|, write Oi(u)={z1,z2},L(v) ={P,Q}, and assume without loss of generality that P∩z1=∅and Q∩z2=∅as subsets of R2. Then, we define the bijection μi(u):Oi(u)→L(v) by declaring μi(u)(z1)=P, μi(u)(z2)=Q(see Fig. 8). The next result is a consequence of Lemma 4.12. Lemma 4.14 Let u ivjw,ujviw. Then, the following squares commute:
95 Page 20 of 45 F. C. Morán, M. Silvero (1) If (u)=(v)=0and (v) =(w) =1, then1 Oi(u)μi(u) hu,v L(v) ρv,w Oi(v)μi(v)L(w). (2) If (u)=0and (v) =(v)=(w) =1, then Oj(u)=Oi(u)μi(u) μj(u) L(v) ρv,w L(v) ρv,w L(w). 5 Khovanov functors In this section we review the functor given by Lipshitz and Sarkar [5,6] and introduce a new Khovanov functor, giving a natural transformation between them which allows us to prove that the geometric realizations of both functors are homotopy equivalent at the almost-extreme quantum grading (Corollary 5.7). First, we introduce some notation. Let ube a state of a diagram Dso that (u)=0. Orient G(u)by fixing, for each of its edges, one of its two possible orientations. We define, for each e∈E(u),the subgraph e+⊂G(u)as the connected component of G(u){e}towards which eis pointing. Note that e+is well defined, since Lemma 4.3 guarantees that G(u)has no cycles. Given any two states u>vof D, there is an inclusion of the associated graphs G(u)⊂G(v), and we define the maps fu,v :π0G(u)π0G(v), gu,v :E(u)→E(v);(5.1) observe that the first one is an isomorphism if (u)=(v), while the second one is always injective. Additionally, if uiv, there exists a Burnside morphism hu,v :Z(u)−→ Z(v) mapping a circle z∈Z(u)either to itself if z/∈Oi(u),ortoOi(v) if z∈Oi(u). 5.1 The Khovanov functor in almost-maximal grading The key piece in the construction of the Khovanov spectra in [6]wastheKhovanov functor, whose associated stable homotopy type is a link invariant. Using Lemma 1We defer the definition of hu,v :Z(u)−→ Z(v) some lines until the beginning of Sect. 5.
Almost-extreme Khovanov spectra Page 21 of 45 95 3.2, we restate now this functor, adapted to the particular case of the almost-maximal quantum grading. Given a link diagram Dwith nordered crossings, consider the functor2 F:2n−→ B defined, in a vertex u∈2n,as 3: F(u)=⎧ ⎪ ⎨ ⎪ ⎩ Z(u)if (u)=0, {u+}if (u)=1, ∅if (u)>1. On a morphism ϕu,v, with uiv, the span Fu>v := F(ϕu,v):F(u)→F(v) is given, for z∈Z(u)and u+=(u,x+),by Fu>v(z)=⎧ ⎪ ⎨ ⎪ ⎩ hu,v(z)if (v) =0, v+if (v) =1andz∈Oi(u), ∅if (v) =1andz/∈Oi(u); Fu>v(u+)=v+if (v) =1, ∅if (v) =2. Finally, we specify 2-morphisms: Let uivjwand ujviw.We need to produce a 2-morphism Fu,v,v,w between the 1-morphisms Fv>w ◦Fu>v and Fv>w ◦Fu>v. Consider first the case when (u)=0. If (u)=(v) =(v)=0, (w) =1, then the chords ei(u)and ej(u)form an alternating pair attached to some circle z1,so Oj(v) =Oi(v) and Oi(v)=Oj(v).Ifz∈Z(u)and z= z1, then Fv>w ◦Fu>v(z)= ∅. In the case z=z1, label the circles involved in the mergings and splittings as {z1}=Oi(u)=Oj(u), {z2,z3}=Oi(v) =Oj(v), {z6}=Oi(w) =Oj(w), {z4,z5}=Oi(v)=Oj(v). Therefore, Fv>w ◦Fu>v(z1)=Fv>w(z2+z3)={z2}·w++{z3}·w+=Oj(v) ·w+, Fv>w ◦Fu>v(z1)=Fv>w(z4+z5)={z4}·w++{z5}·w+=Oi(v)·w+. 2When working with different diagrams, we write FDto denote the functor Fassociated to the link diagram D. 3The original functor Ffrom [6] splits into functors Fjwhich associates to each state uasetEu,jof enhancements so that q(u,x)=j,foreveryx∈Eu,j. When particularizing to the case when j=jalmax , Corollary 4.5 implies that if (u)=0, then the set Eu,jcan be identified with Z(u), whereas if (u)=1 then Eu,j={u+}.
95 Page 22 of 45 F. C. Morán, M. Silvero Then, the 2-morphism Fu,v,v,w consists of a bijection between Oj(v) and Oi(v) given by the ladybug matching μi(v)−1◦μj(v) :Oj(v) ↔L(w) ↔Oi(v)(see Sect. 4.6). If (u)=0 and we are not in the previous situation, then every summand in the formal sums (Fv>w ◦Fu>v)(z)and (Fv>w ◦Fu>v)(z)has a singleton as coefficient, so there is a unique choice for the 2-morphism Fu,v,v,w. Inthecasewhen(u)=1,then Fv,w◦Fu,v and Fv,w ◦Fu,vareemptyif(w) > 1; otherwise, both Fv,w ◦Fu,v and Fv,w ◦Fu,vmap u+to the formal sum w+which has a singleton as coefficient. Lemma 5.1 [5]F satisfies conditions (C1) and (C2) in Lemma 3.2. The (strictly unitary lax) 2-functor Fdefined above is the Khovanov functor given in [6] at the almost-maximal quantum grading and, if Dhas n−negative crossings, the almost-extreme (maximal) Khovanov spectrum is defined as Xjalmax D=−n−∞Tot FD.(5.2) 5.2 A new equivalent functor We introduce now a (strictly unitary lax) 2-functor M:2n→B, with the property that its realization coincides with the realization of F, as will be show in Corollary 5.7. As before, we start by defining4Mfor the vertices of the cube, corresponding to the states of D: M(u)=⎧ ⎪ ⎨ ⎪ ⎩ π0G(u)∪E(u)if (u)=0, u+if (u)=1, ∅if (u)>1. On a morphism ϕu,v, with uiv, the span Mu>v := M(ϕu,v):M(u)→M(v) is defined either on a connected component Cor an edge eof G(u)(if (u)=0), or on the enhanced state u+(if (u)=1), as follows: Mu>v(C)=fu,v(C)if (v) =0, (L(v) ∩fu,v(C)) ·v+if (v) =1; Mu>v(e)=gu,v(e)if (v) =0, (μi(u)(e+∩Oi(u)) ·v+if (v) =1; Mu>v(u+)=v+if (v) =1, ∅if (v) =2. Remark 5.2 Observe that the expression L(v)∩fu,v(C)equals either L(v), in the case when Oi(u)∩C=∅, or the empty set otherwise. 4When working with different diagrams, we write MDto denote the functor Massociated to the link diagram D.
Almost-extreme Khovanov spectra Page 23 of 45 95 Fig. 9 We illustrate the three cases considered in Remark 5.3.GivenD(u)and the orientation shown in G(u),e+∩O1(u)=∅,e+∩O2(u)={z2}and e+∩O3(u)={z1,z2} Fig. 10 Cases (a), (b)and(c) in the definition of Mu,v,v,w.Inorder|L(w)|=2, |π0G(w)|must be equal to |π0G(1)|and therefore the vertices corresponding to the two circles depicted in D(u)in cases (b)and (c)must belong to the same connected component in G(u) Remark 5.3 Observe that the evaluation of the expression e+∩Oi(u)leads to either the empty set or one or two circles, as illustrated in Fig. 9. When e+∩Oi(u)=z∈ Z(u), there are two possibilities: if |π0G(v)|=|π0G(1)|, then μi(u)(e+∩Oi(u)) maps zto one of the two arcs in L(v); however, if |π0G(v)|=|π0G(1)|−1, then μi(u)(e+∩Oi(u)) maps zto the singleton L(v). Finally, given uivjwand ujviw, we define the 2-morphism Mu,v,v,w between the 1-morphisms Mv>w ◦Mu>v and Mv>w ◦Mu>vas follows. First, note that Mu,v,v,w is trivially defined when (u)=(v) or |L(w)|=1, since it is a bijection between singletons. Then, we just need to specify the cases when (u)=0, (w) =1 and |L(w)|=2, depicted in Fig. 10. Moreover, the value of Mv>w ◦Mu>v and Mv>w ◦Mu>von a component C∈π0G(u)equals the empty set or a singleton, unless fu,w(C)is the component of G(w) containing two cycles; a similar reasoning applies for edges e∈G(u), unless e(or possibly gu,v(e)or gu,v(e)) points to the two circles involved in the unique merging. We define Mu,v,v,w in such components and edges: (a)If(u)=(v) =(v)=0, (w) =1 and |L(w)|=2, then ei(u)and ej(u) form an alternating pair (see Fig. 10a) and we have Mv>w ◦Mu>v(C)=L(w) ·w+ Id ←→ Mv>w ◦Mu>v(C)=L(w) ·w+, Mv>w ◦Mu>v(e)=L(w) ·w+ Id ←→ Mv>w ◦Mu>v(e)=L(w) ·w+.
95 Page 24 of 45 F. C. Morán, M. Silvero (a) (b) (c) (d) (e) (f) (g) Fig. 11 Moving along the cube. Paths (a),(b)and (c)represent M001>000 ◦M011>001 ◦M111>011, M010>000 ◦M011>010 ◦M111>011 and M010>000 ◦M110>010 ◦M111>110, respectively (b)If(u)=(v)=0 and (v) =(w) =1 and |L(w)|=2, then ei(u)is a bichord and ej(u)a monochord (see Fig. 10b) and we have Mv>w ◦Mu>v(C)=L(v) ·w+ ρv,w ←→ Mv>w ◦Mu>v(C)=L(w) ·w+, Mv>w ◦Mu>v(e)=L(v) ·w+ ρv,w ←→ Mv>w ◦Mu>v(e)=L(w) ·w+. (c)If(u)=0 and (v) =(v)=(w) =1 and |L(w)|=2, then ei(w) and ej(w) are parallel bichords (see Fig. 10c) and we have: Mv>w ◦Mu>v(C)=L(v) ·w+ ρ−1 v,wρv,w ←→ Mv>w ◦Mu>v(C)=L(v)·w+, Mv>w ◦Mu>v(e)=L(v) ·w+ ρ−1 v,wρv,w ←→ Mv>w ◦Mu>v(e)=L(v)·w+. Lemma 5.4 The 2-functor M satisfies conditions (C1) and (C2) in Lemma 3.2 and therefore is well defined. Proof Condition (C1) follows from the definition of M. Consider an arbitrary threedimensional face and denote its vertices as follows: 111 M111>011 011 M011>001 101 001 M001>000 110 010 100 000. (5.3) In order to prove condition (C2), we need to show that 2-morphisms in this cube commute. To do so, we study the bijections obtained when we move along the cube. More precisely, starting from M001>000 ◦M011>001 ◦M111>011 we move to M010>000 ◦ M011>010 ◦M111>011 and continue moving along the cube following Fig. 11 until we reach again M001>000 ◦M011>001 ◦M111>011. To simplify notation, write ρifor the morphism ρu,v if uiv; then, the commutative diagram of Lemma 4.13 associated to uivjwand ujviw becomes:
Almost-extreme Khovanov spectra Page 25 of 45 95 L(u)ρi ρj L(v) ρj L(v)ρiL(w). (5.4) We just need to study the cases when (111)=0, (000)=1 and |L(000)|=2 (otherwise,thecoefficientsinvolvedinallcompositionsof1-morphismsoftheedgesof thecubearesingletons,andthereforethe2-morphismscommutetrivially).Inprinciple, up to permutation, there are 8 different situations attending to the value of in each vertex, depicted in Fig. 12. However, it is not hard to check that situations (a),(e)and (f)leads to inconsistency in the chord diagrams, and therefore they are not possible. We study the remaining 5 possible situations: (a) Figure 12b (i.e., (110)=(101)=(011)=(100)=(010)= 0,(001)=1): D(111)000 contains three monochords, two of them constituting an alternating pair. Coefficients involved in the three-fold compositions when moving around the cube are either singletons or the following: L(001)ρ3 ←→ L(000)=L(000)=L(000)=L(000)ρ−1 3 ←→ L(001)=L(001). The composition of the 2-morphisms connecting coefficients is the identity morphism. (b) Figure 12c (i.e., (110)=(101)=(011)=(100)=0,(001)= (010)=1): D(111)000 contains three monochords constituting two alternating pairs. Coefficients involved in the three-fold compositions when moving around the cube are either singletons or the following: L(001)ρ−1 2ρ3 ←→ L(010)=L(010)ρ2 ←→ L(000)=L(000)ρ−1 3 ←→ L(001)=L(001). The composition of the 2-morphisms connecting coefficients is the identity morphism. (c) Figure 12d: D(111)000 contains two monochords not forming an alternating pair and a bichord. Coefficients involved in the three-fold compositions when moving around the cube are either singletons or the following: L(011)=L(011)ρ3 ←→ L(010)ρ2 ←→ L(000)=L(000)ρ−1 3 ←→ L(001)ρ−1 2 ←→ L(011).
95 Page 32 of 45 F. C. Morán, M. Silvero Definition 7.2 The categorical dual of a downwards (upwards) closed subposet X⊂ 2nis the upwards (downwards) closed subposet X∗⊂2n→Spgiven by u∈X∗if and only if ¯u∈X. The complement of a downwards (upwards) closed subposet X⊂2nis the upwards (downwards) closed subposet ˆ X⊂2ngiven by u∈ˆ Xif and only if u/∈X. The complement of the categorical dual of a downwards closed set Xis again a downwards closed subposet of the cube whose associated simplicial complex is the Alexander dual of X. In general |ˆ X|≃|X|if Xis downwards closed and |X∗|is Spanier-Whitehead dual to |X|. Observe that (1) If |X|≃|A|∨|B|, then |X∗|≃|A∗|∨|B∗|, (2) If |X|≃Sk, then |X∗|≃Sn−k−2, where nis the dimension of the cube. Given a link diagram D, the upwards closed subposet of the cube given by those states usuch that (u)=0, which we denote XD, corresponds to the functor Fjmax :2n→Set•⊂B. Corollary 7.3 XDis the categorical dual of the poset of faces of ID. As a consequence if |ID|≃Sk, then |XD|≃Sn−k−2. 7.2 Decomposing MD Given a state uof a diagram D, recall that we write D(1)ufor the chord diagram having the same circles as D(1)and those chords ei(1)so that ui= 1. Given a crossing cin D, we write D[c=0](resp. D[c=1]) for the link diagram obtained after smoothing cfollowing a 0-label (resp. 1-label). Definition 7.4 Given a link diagram Dwith ncrossings, we consider the following subposets of the cube 2n: •XD: is the subposet of 2nconsisting of those states uso that (u)=0 (as defined in previous section). •Xe D=XD[e=0], for any monochord e∈D(1). •YD: is the subposet of 2nconsisting of those states uso that (u)=1 and D(1)u contains an alternating pair. •Zb D: is the subposet of 2nconsisting of those states uso that (u)=1 and D(1)u contains a bichord parallel to a given bichord bof D(1). Observe that each state (admitting an enhacement) in the almost-extreme complex of Dbelongs to one and only one of the previous subcubes XD,YDor ZD(see Corollary 4.5 and Proposition 4.8). Proposition 7.5 The functor MDcan be realized as the following cofibre sequence −1⎛ ⎝ Z(1) XD∨ e∈N Xe D⎞ ⎠−→ YD∨ [b]∈B Zb D−→ MD,(7.1) where N is the set of monochords in G(1)and B the set of classes of parallel bichords.
Almost-extreme Khovanov spectra Page 33 of 45 95 The decomposition follows because the second map is levelwise injective and the suspension of the leftmost term is precisely the quotient functor of this second map. Remark 7.6 A Mayer-Vietoris spectral sequence allows to compute both posets YD and Zb Din terms of some posets XD, where Dare partially smoothed link diagrams of D. We explain this below. The poset YDcan be covered with the subposets XD[e=0,f=0]where {e,f}runs along the set of all alternating pairs in D(1).Thek-fold intersections Akof this covering are Ak= {e1,...,ek+1} XD[e1=0,...,ek+1=0], where {e1,...,ek+1}are monochords attached to the same circle in such a way that they can be divided into two non-empty sets H1and H2so that all monochords in H1 alternate with all monochords in H2and viceversa. Similarly, the poset Zb Dcan be covered with the subposets XD[b=0]where bruns along the set of bichords of D(1), and the k-fold intersections Bkof this covering are Bk= {b1,...,bk} XD[b1=0,...,bk=0], where {b1,...,bk}are bichords parallel to b. Thus, the (first page of the) Mayer-Vietoris spectral sequences associated to these coverings are: Hp⎛ ⎝ {e0,...,eq+1} XD[e0=0,...,eq+1=0]⎞ ⎠⇒Hp+q(YD), Hp⎛ ⎝ {b0,...,bq} XD[b0=0,...,bq=0]⎞ ⎠⇒Hp+q(Zb D). Example 7.7 Consider the torus knot T(3,q)and let D=D(3,q)be its standard diagram. Since D(1)contains no bichords, Zb Dis trivial. We will combine Remark 7.6 together with Corollary 7.3 to compute the realization of subposets in (7.1). More precisely, we will express XD,Xe Dand YD(via Ak) as the duals of independence complexes of some Lando graphs. Write Cnfor the cycle graph of nvertices and Ln the path of length n. –GL(D)=C2q; –GL(D[e=0])=L2q−4, for any monochord ein D(1); –GL(D[e1=0,e2=0])=C2q−2whene1ande2aretwoconsecutivemonochords in D(1). Note that alternating pairs consist precisely in consecutive monochords;
95 Page 34 of 45 F. C. Morán, M. Silvero –GL(D[e1=0,e2=0,e3=0])=L2q−6when e1,e2,e3are three consecutive monochords in D(1). Note that the previous condition is equivalent to require that e1,e2,e3can be divided into two subsets H1and H2such that all monochords in each subset alternate with all monochords in the other subset. Note that there are no n-tuples of monochords e1,...,enwhich can be divided into two subsets H1and H2satisfying the condition above when n>3. Summarizing we have: XD=I∗ C2qXe D=I∗ L2q−4Ak=⎧ ⎪ ⎨ ⎪ ⎩ 2qI∗ C2q−2if k=1, 2qI∗ L2q−6if k=2, ∅if k>2. The homotopy types of the above complexes were computed in [10, Corollary 3.4 and Proposition 3.9]: |ICn|≃Sk−1if n=k±1, Sk−1∨Sk−1if n=3k.|ILn|≃Skif n=3k+1,3k+2, ∗if n=3k. This allows to compute the almost-extreme Khovanov spectra of T(3,q)in terms of well-known independence complexes of cycles and paths, with no need to apply induction on q. Remark 7.8 The cone-length c(X)of a spectrum Xis the least nsuch that there is a sequence of cofibre sequences Yi→Xi→Xi+1 for i=0,...,n−1, such that X0is contractible, Yiis a wedge of spheres and Xn=X.In[10] it was conjectured that |XD|is homotopy equivalent to a wedge of spheres for any diagram D. If this were true, then Remark 7.6 would imply that the cone length of Tot MDis bounded above by the maximum of the following numbers: (1) the maximum number of parallel bichords plus one in D(1). (2) the maximum number of monochords in D(1)that can be partitioned into two disjoint subsets H1,H2such that all chords in H1alternate with all chords in H2 and viceversa. 7.3 Skein sequences Let abe a crossing in a link diagram D. Observe that if ais a monochord in D(1)we get the skein short exact sequence7 MD[a=1]−→ MD[a=0]−→ MD,(7.2) 7Compare to skein short exact sequences from [13].
Almost-extreme Khovanov spectra Page 35 of 45 95 whereas if ais a bichord in D(1), then the skein short exact sequence becomes MD[a=1]−→ XD[a=0]−→ MD.(7.3) Moreover, when ais a monochord in D(1)we also have the following sequence: XD[a=1]−→ XD[a=0]−→ XD.(7.4) To verify the three above sequences note that the second map is an inclusion whose quotient is the suspension of the first spectrum. Definition 7.9 Let Dbe a diagram and let abe a monochord in D(1). We say that a is: (1) 2-free if it is not part of any alternating pair in D(1). (2) 3-free if it is not part of any alternating triple in D(1). (3) free if it is 2-free and 3-free. (4) b-free if it is not part of any alternating triple involving the bichord bin D(1). Lemma 7.10 Let D be a diagram and let a be a monochord in D(1). (1) If a is 2-free, then X D,YDand all Xe D, with e = a a monochord, are contractible; if, additionally, D(1)contains another 2-free monochord, then Xa Dis contractible too. (2) If a does not form an alternating triple with a bichord b (i.e., a is b-free), then Zb D is contractible. Proof The leftmost map in skein sequence (7.4) along the crossing associated to ais identity, hence XDis contractible. A similar reasoning works for Xe Dwhen e= a. In general, the skein sequence (7.2) does not respect the decomposition in (7.1); however, when ais 2-free, it respects the term YDand when ais b-free it respects Zb D. In both cases the maps YD[a=1]→YD[a=0]and Zb D[a=1]→Zb D[a=0]are the identity. Corollary 7.11 If D(1)contains at least two 2-free monochords, then MD≃ [b]∈B Zb D. Proof The result is direct from cofibre sequence (7.1) and Lemma 7.10(1). Corollary 7.12 If D has a free monochord e, then MD≃XD[e=0]. Proof Lemma 7.10 implies that the only possibly non-contractible subposet in (7.1) is Xe D=XD[e=0]. Lemma 7.13 Let a be a 2-free monochord in D(1)dividing a circle into two regions so that both of them contain at least one 2-free monochord (we say that D(1)contains nested monochords). Then MD≃MD[a=1].
95 Page 36 of 45 F. C. Morán, M. Silvero Proof We use skein sequence (7.2) and show that MD[a=0]is contractible. First, notice that the circle in the statement is split into two circles z1and z2when changing the 1-label of afor a 0-label, each of them attached to at least a 2-free monochord that we call cand d, respectively. Lemma 7.10(1) implies that the the only possibly noncontractible spaces when applying cofibre sequence (7.1)toD[a=0]are Zb D[a=0]. Now, since ais 2-free in D(1), there is no bichord connecting z1and z2in D[a= 0](1). Therefore, for any bichord bin D[a=0](1)at least one of the monochords c or dare b-free, so Lemma 7.10(2) implies that Zb D[a=0]is contractible for all bichords b, thus MD[a=0]is contractible. Definition 7.14 Two parallel bichords aand bof a chord diagram D(1)are equivalent if there is no monochord eso that a,band econstitute an alternating triple. Lemma 7.15 If D has two equivalent bichords a,b, then MD≃MD[a=1]. Proof Since aand bare equivalent, the chord bis 2-free in the chord diagram D[a= 0](1), so Lemma 7.10(1) implies that XD[a=0]is contractible. The statement follows from the skein sequence (7.3) along the chord a. 8 Diagrams with no alternating pairs In this section we determine the homotopy type of MD(thus, that of the Khovanov spectrum given by Lipshitz and Sarkar in [7] for the almost-extreme quantum grading) for diagrams Dso that D(1)contains no alternating pairs. Observe that this determines their Khovanov homology groups at almost-extreme quantum degree. 8.1 Diagrams with no monochords This case was studied in [11]: If Dis 1-adequate (i.e., D(1)has no monochords), then MDis homotopy equivalent to a wedge of spheres and possibly a suspension of the projective plane. See [11, Theorem 1.1] for the details. 8.2 Diagrams with one monochord Proposition 8.1 Let D be a link diagram with n crossings so that D(1)contains a single monochord attached to a circle z, and let k be the number of circles connected to z along an alternating triple. Then, |MD|≃ k−1 Sn−3. Proof Write efor the monochord in D(1). By Lemma 7.10(1), the decomposition in (7.1) becomes −1Xe D−→ [b]∈B Zb D−→ MD.
Almost-extreme Khovanov spectra Page 37 of 45 95 The subposet Xe Dhas a single element in degree n−1, thus |Xe D|≃Sn−2. By Lemma 7.10(2), Zb Dis contractible unless bforms an alternating triple with e. Therefore we mayassume,uptosuspension,thatallbichordsareattachedtozandformanalternating triple with e. Moreover, using Lemma 7.15 we can assume, up to suspension, that circles are connected to zby exactly two bichords. Therefore, for each such bichord b, the subposet Zb Dhas five elements in M-shape, with two elements in degree n−1andthreeindegreen−2, thus |Zb D|≃Sn−3, and the above decomposition becomes Sn−3−→ [b]∈B Sn−3−→ | MD|. The map to the wedge of spheres is a diagonal map. This concludes the proof. 8.3 Diagrams with 2-free monochords Let Dbe a diagram whose associated chord diagram D(1)contains more than one monochord, all of them 2-free. Recall from Corollary 7.11 that in this situation MD≃ [b]∈BZb D, thus we can restrict to the independent study of each connected pair of circles when computing Zb D. At this point we can make some simplifications in D(1): Lemmas 7.13 and 7.15 allow us to remove nested monochords and equivalent bichords when computing MD (in exchange of taking suspensions). Moreover, by Lemma 7.10(2) we can assume that D(1)contains no b-free monochords for any bichord b. These simplifications motivate the following definition: Definition 8.2 A diagram Dis simple if the associated chord diagram D(1)contains at least two monochords, all of them 2-free, and satisfies the following conditions: (1) It contains exactly two circles; (2) It contains no nested monochords; (3) There are no b-free monochords for any bichord b; (4) There are no equivalent bichords. By definition, if Dis simple then D(1)can be isotoped in S2so that monochords and bichords lie in different regions (see Fig. 13). Moreover, since D(1)contains no nested monochords, a circle with nmonochords is divided into nhalf-disks (each of them bounded by a monochord and an arc of the circle) and an additional region (bounded by the nmonochords and narcs of the circle) that we call polygon. Given a monochord ein D(1), we write defor the half-disk bounded by e8, and |de| for the number of bichords having one of their endpoints in de. In simple diagrams |de|≥1 for any monochord e. In the proofs of Lemmas 8.3,8.7 and 8.8 we use several results from [10, Section 3]. 8If eis the only monochord attached to the circle, then there are two possible options for de; in this case choose an option minimizing |de|, without loss of generality.
95 Page 38 of 45 F. C. Morán, M. Silvero (a) (b) Fig. 13 Two chord diagrams corresponding to simple diagrams. Grey regions correspond to half-disks bounded by monochords. In a|de|=3, while |de|=1 for every monochord ein (b) (a) (b) (c) Fig. 14 The 1-resolutions corresponding to D,D[b=0]and D[b=0,e=0]illustrating the proof of Lemma 8.3 are shown in (a–c), respectively. Regions Rand Rare shaded green and grey, respectively (color figure online) Lemma 8.3 Let D be a simple diagram whose associated chord diagram D(1)contains a monochord e so that |de|=1. Then |MD|is either contractible or homotopy equivalent to a sphere. Proof Let bbe the only bichord with an endpoint in deand consider the skein sequence (7.3) along b: MD[b=1]−→ XD[b=0]−→ MD. The monochord eis free in D[b=1](1), thus by Lemma 7.10 ZD[b=1]is contractible, thus MD[b=1]is contractible (by Corollary 7.11) and MD≃XD[b=0]. Now, D[b=0](1)has a single circle and its inner part is separated into two regions Rand Rby the monochord e. See Fig. 14b. Moreover: (1) monochords in D(1)become internal monochords in D[b=0](1); (2) monochords attached to each of the circles in D(1)become monochords contained in each of the regions R,Rin D[b=0](1); (3) bichords in D(1)become external monochords in D[b=0](1)having one endpoint in Rand the other one in R.
Almost-extreme Khovanov spectra Page 39 of 45 95 Consider now the skein sequence (7.4) along the monochord e XD[b=0,e=1]−→ XD[b=0,e=0]−→ XD[b=0]. Condition (2) above implies that there are no alternating pairs in D[b=0,e=0](1) (see Fig. 14c), so the middle term in the above sequence is contractible and we have XD[b=0]≃XD[b=0,e=1]. Assume first that bconnects dewith the region that we call polygon, then we claim that the Lando graph Gof D[b=0,e=1], contains no cycles (i.e., it is a forest). In order to prove that, we proceed by contradiction. Suppose that Cnis a cycle of length nin G, with vertices e1,e2,...,ennumbered so that e1is an internal chord in the region R. Conditions (1) and (2) imply that monochords e4k+1and e4k+3lie in Rand R, respectively. Moreover, the absence of equivalent bichords in D(1)implies the absence of cycles of length four in G, i.e., n≥8. The monochord e4separates the external region into two subregions S,S. Since there are no cycles of length 4, e2and e6lie one in each of these two subregions, say e2∈Sand e6∈S. Moreover, as the internal monochords are non-nested, all internal monochords but e3and e5have both endpoints in either Sor S. Therefore (1) the endpoints of e1lie in S(because e2lies in S) and (2) the endpoints of e2k+1lie in Sif k≥3 (because e2klies in Sif k≥3). In particular, both endpoints of en−1lie in S. This is a contradiction, since enis a external monochord connecting e1and en−1. Therefore Gcontains no cycles, and by [10, Corollary 3.7] its associated independence complex is either contractible or a sphere. Corollary 7.3 completes the proof. Assume now that bhas its endpoints in two half-discs deand df, for some monochord f, and consider the skein sequence (7.4) along f. We get that XD[b=0,e=1,f=0] is contractible, since all monochords in Rare 2-free. Therefore, XD[b=0,e=1]≃ XD[b=0,e=1,f=1]. The latter is the same as XD[b=0,e=1], where D=D[f=1]is a diagram where econnects dewith the polygon, which was studied in the previous case. Remark 8.4 Since simple diagrams do not contain equivalent bichords, if |de|≥4for some monochord e, then there exists a monochord fwith |df|=1(dfis connected to deby a bichord). In other words, if |de|>1 for all monochords e, then |de|≤3for all monochords e. Definition 8.5 A simple diagram Dso that 2 ≤|de|≤3 for every monochord eof D(1)is called super-simple. If Dis super-simple, every half-disk dein D(1)satisfies one of the three following conditions: (1) |de|=2, and both bichords connect dewith two contiguous half-disks in the other circle (see Fig. 15a); (2) |de|=2, and one bichord connects dewith a half-disk in the other circle and the other one connects dewith the polygon in the other circle (see Fig. 15b);
95 Page 40 of 45 F. C. Morán, M. Silvero (a) (b) (c) (d) Fig. 15 The three possible situations for a half-disk dein the 1-resolution of a super-simple diagram are illustrated in figures (a–c). Situation dis not allowed in simple diagrams (a) (b) (c) Fig. 16 The 1-resolutions corresponding to D,D[a=0,f=1]and D[a=0,f=0]illustrating the proof of Lemma 8.7 are shown in (a–c), respectively. Dotted lines represent those chords that may or may not be in the diagrams (3) |de|=3, and two bichords connect dewith two contiguous half-disks in the other circle, and the third one is placed between them and connects dewith the polygon of the other circle (see Fig. 15c). The next statement follows from Lemmas 8.7 and 8.8. Proposition 8.6 Let D be a super-simple diagram. Then |MD|is homotopy equivalent to a wedge of spheres. Lemma 8.7 Let D be a super-simple diagram so that D(1)contains at least one bichord connecting a half-disk with the region called polygon. Then |MD|is either contractible or homotopy equivalent to a sphere. Proof Let bbe a bichord connecting a half-disk dein a circle z1with the polygon of the circle z2, for some monochord e. See Fig. 16a. Since Dis super-simple, there exists at least a bichord aconnecting dewith dffor a monochord fattached to z2. We distinguish two cases, depending on whether z2contains at least two monochords or it contains just a single monochord.
Almost-extreme Khovanov spectra Page 41 of 45 95 Case 1: Assume that the chord diagram D(1)contains more than one monochord attached to z2, and consider the skein exact sequence (7.3) along a: MD[a=1]−→ XD[a=0]−→ MD. We will show that XD[a=0]is contractible. To do so, we apply the skein sequence (7.4) along the monochord f: XD[a=0,f=1]−→ XD[a=0,f=0]−→ XD[a=0]. Figure16brepresents D[a=0,f=1](1), wherebisafree monochord.Moreover, any monochord different from fattached to z2becomes 2-free in D[a=0,f= 0](1), as illustrated in Figure 16c. Therefore by Lemma 7.10(1) XD[a=0,f=1]and XD[a=0,f=0]are contractible, and so is XD[a=0]. As a consequence, MD≃MD[a=1].If|de|=2inD(1), then |de|=1in D[a=1](1), and the statment holds by Lemma 8.3.If|de|=3inD(1), then |de|=2 in D[a=1](1)and it contains a bichord cconnecting dewith dhfor a monochord h (contiguous to f) attached to the circle z2. Applying the skein exact sequence (7.3) along cand repeating the same reasoning as before leads to MD≃2MD[a=1,c=1]. Since |de|=1inD[a=1,c=1](1), Lemma 8.3 completes the proof for this case. Case 2: Assume that there is just one monochord fattached to the circle z2in D(1). Then, since Dis super-simple, D(1)is as one of the six chord diagrams depicted in Fig. 17 (each picture leads to two possible chord diagrams, depending on whether it contains the bichord por not). In order to compute MDfor each of these situations, we use skein sequence (7.3) together with Corollary 7.3.Wealsouse[10, Lemma 3.2, Corollary 3.3, Corollary 3.4] to compute ID: •If D(1)is as in Fig. 17a with p: we consider the skein sequence (7.3) along the bichord c. Since the Lando graph GL(D[c=0])is a path of length four, we deducefrom [10,Corollary 3.4]that|ID[c=0]|≃S1,and therefore|XD[c=0]|≃S2. Next, we show that MD[c=1]is contractible by applying the skein sequence (7.3)to D[c=1]along the bichord a(we use the fact that chord fis free in D[c=1,a=1](1) and use [10] to compute ID[c=1,a=0]). Therefore, |MD|≃|XD[c=0]|≃S2. •If D(1)is as in Fig. 17a without p: we proceed as in the previous case, and get that both MD[c=1]and XD[c=0]are contractible, and so is MD. •If D(1)is as in Fig. 17b with p: we consider the skein sequence (7.3) along the bichord a. First, we compute |ID[a=0]|≃S1, and therefore |XD[a=0]|≃S4.Next, we show that MD[a=1]is contractible by applying the skein sequence (7.3)to D[a=1]along the bichord b(we use the fact that chord eis free in D[a=1,b=1](1) and that ID[a=1,b=0]is contractible). Therefore, |MD|≃|XD[a=0]|≃S4. •If D(1)is as in Fig. 17b without p: the procedure is analogous to the previous one, the only difference is that |XD[a=0]|≃S3and therefore |MD|≃S3. •If D(1)is as in Fig. 17c with p: we consider the skein sequence (7.3) along the bichorda. Since ID[a=0]iscontractible,so is XD[a=0].Next,weshow that MD[a=1] is also contractible by applying the skein sequence (7.3)toD[a=1]along the bichord b(we use the fact that chord eis free in D[a=1,b=1](1)and that ID[a=1,b=0] is contractible). Therefore, MDis contractible.