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Direct computation of Khovanov homology and knot Floer homology

Taketo Sano
October 20, 2019

Direct computation of Khovanov homology and knot Floer homology

Workshop "Topology and Computer 2019"
http://www.libe.nara-k.ac.jp/~han/topology_comp_2019_e.htm

Taketo Sano

October 20, 2019
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  1. 1/34 Direct computation of Khovanov homology and knot Floer homology

    Taketo Sano The University of Tokyo 2019-10-20 Topology and Computer 2019
  2. 2/34 Overview knot homology theory Khovanov homology(’00) HKh(K) HFK(K) knot

    Floer homology(’02) knot concordance invariant Rasmussen invariant (’04) s(K) ∈ 2Z Tau invariant (’02) τ(K) ∈ Z Upsilon invariant (’14) ΥK (t) ∈ PL([0, 2], R) 1 2 3 4
  3. 3/34 In this talk, I will describe algorithms for computing:

    1. Khovanov homology and the s-invariant, 2. Knot Floer homology and related invariants. The two topics are described in parallel. The former is simpler in both theory and computation. The algorithms for the latter are results of a joint work with Kouki Sato. Main Result We obtained an explicit algorithm for computing the Υ-invariant of knots. As an application, we have determined Υ for almost all (except for 4) knots with crossing number up to 11, including 39 knots whose Υ have been unknown. For each of the missing 4 knots, we have narrowed the candidates up to 3. ∗ The paper is currently in progress.
  4. 5/34 Khovanov homology Khovanov homology HKh is a bigraded link

    homology theory, constructed combinatorially from a link diagram. It is a is a categorification of the Jones polynomial, i.e. its graded Euler characteristic of gives the (unnormalized) Jones polynomial. Figure 1: Khovanov homology for K = 01, 31, 41
  5. 6/34 Construction of the chain complex (sketch) Let D be

    a link diagram with n crossings. Consider all possible choice of simultaneous resolutions of the crossings (there are 2n), each of which yield a diagram consisting of disjoint union of circles. Place them on the vertices of an n-dimensional cube, so that each edge corresponds to a single resolution change. Transform the cube into a commutative diagram in the category of R-modules, where R is an arbitrary commutative ring. After appropriately modify the signs on the edges, fold the cube into a sequence and we obtain the chain complex CKh(D; R).
  6. 7/34 Computation of Khovanov homology If R is a (computable)

    field such as Q or Fp, every calculations involved are linear maps between finite dimensional vector spaces, hence the homology group can be computed directly. More generally, if R is a (computable) Euclidean domain such as Z or Q[t], then the computation is possible by the following method: Proposition (The Smith Normal Form = SNF) If R is a PID, any matrix A can be transformed uniquely (up to units) into the following form: A ∼ diag(1, · · · , 1 p , a1, · · · , aq q , 0, · · · , 0 r ) where a1 ∼ 1 and a1 | · · · | aq. In particular if R is an Euclidean domain, there is an explicit algorithm for computing the form.
  7. 8/34 The algorithm Step 1. For each i, take the

    basis of the i-th chain module Ci consisting of tensor products of 1s and Xs. Step 2. Represent the differential di : Ci → Ci+1 by a matrix Di . Step 3. Compute bases for the cycle module Zi (from the solution space of Di x = 0), and the boundary module Bi (from the linearly independent columns of Di−1). Step 4. Since Bi = bi 1 , · · · , bi l is a submodule of Zi = zi 1 , · · · , zi k , there is a matrix T such that (bi 1 , · · · , bi l ) = (zi 1 , · · · , zi k )T. Step 5. Compute the SNF S of T as in the previous slide, and we obtain the decomposition of the quotient module: Hi = 0 ⊕ · · · ⊕ 0 p ⊕ R/(a1) ⊕ · · · ⊕ R/(aq) q ⊕ R ⊕ · · · ⊕ R r .
  8. 10/34 Lee homology and the filtered q-degree Lee homology HLee

    is a deformed version of Khovanov homology. Its differential is modified as dLee = dKh + Φ, where Φ is a differential that anticommutes with the original dKh. Because of Φ, the differential d no longer preserves the secondary degree (the quantum degree or q-degree) of CKh. Nevertheless Φ is q-degree ::::::::::::: non-decreasing, so the chain complex admits a filtration: Fj Ci (D) := {x ∈ Ci (D) | qdeg x ≥ j} ⊂ Ci (D) · · · ⊃ Fj−1Ci ⊃ Fj Ci ⊃ Fj+1Ci ⊃ · · · This induces a filtration on the homology by: Fj Hi (D) := Im (Hi (Fj C∗) → Hi (C∗)) ⊂ Hi (D) · · · ⊃ Fj−1Hi ⊃ Fj Hi ⊃ Fj+1Hi ⊃ · · ·
  9. 11/34 So we have [z] ∈ Fj Hi ⇔ ∃z

    ∈ Fj Ci s.t. z ∼ z (homologous). We define the filtered q-degree by qdeg [z] = max{ j | [z] ∈ Fj Hi } i j i0 z i j i0 z qdeg[z] i0 − 1 d
  10. 12/34 Rasmussen’s s-invariant Now suppose R = Q, and D

    is a knot diagram. Lee proved that HLee(D; Q) possesses an explicit basis {[α], [β]}, with both elements lying in homological degree 0. Subsequently, Rasmussen proved: Proposition qdeg [α] and qdeg [β] are equal, and they are invariant under the Reidemeister moves. This justifies the following definition: Definition For a knot K, define: s(K) := qdeg [α] + 1.
  11. 13/34 Computation of s(K) We found that the computation of

    s(K) reduces to considering a sequence of linear systems. First, [α] ∈ Fj H0 is equivalent to: ∃c ∈ C−1 s.t. α − dc ∈ Fj C0. Let Qj C∗ = C∗/Fj C∗, and α be the image of α. Then the above condition is equivalent to: ∃c ∈ Qj C−1 s.t. dc = α ∈ Qj C0 Now each Qj Ci is a finite dimensional Q-vector space. Represent the differential d by a matrix A, then the element c in question corresponds to the solution x of the linear system: Ax = vec(α ).
  12. 14/34 The algorithm Step 0. Compute the cycle α ∈

    C0. Step 1. Assign to j the minimum filtration level of F∗C0 (note that the filtration is bounded). Step 2. Take the basis of Qj C∗ = C∗/Fj C∗ by modding out the generators whose qdeg ≥ j. Step 3. Compute the matrix A corresponding to the differential: d : Qj C−1 → Qj C0. Step 4. Check whether the linear system Ax = vec(α ) has a solution. If it does, increment j (by 2) and goto Step 2. If it doesn’t, then we obtain s(K) = j − 1.
  13. 16/34 Knot Floer homology Knot Floer homology HFK is another

    link homology theory, originated from the Heegaard Floer homology which an invariant of closed 3-manifolds. Although HFK involves heavy analytic machineries, lately, a purely combinatorial description was found. The combinatorial version is also called grid homology in its own right. Figure 2: Grid diagram and the corresponding knot
  14. 17/34 Construction of the chain complex (sketch) Let G be

    a grid diagram of a link, and N be its grid number. The complex C−(G) is a finitely generated free module over F2[U1, · · · , UN], where: the generators {x} are given by permutations of length N. (Each x can be drawn as N-tuple of points on the lattice), the homological degree of x is given by the Maslov function MG (x), with each factor Ui contributing to degree −2, and the differential ∂ : C− k (G) → C− k−1 (G) is given by ∂x = y r∈Recto(x,y) Uε1 1 · · · UεN N y where r runs over the empty rectangles connecting x to y, and the exponents ε1, . . . , εN ∈ {0, 1} are given by counting the number of intersections of r and ’s.
  15. 19/34 Computation of H− ∗ (G) Since the ground ring

    F2[U1, · · · , UN] is not a PID, the methods used for computing Khovanov homology is not applicable. However, since the generators are finite and deg Ui = −2, we may regard each k-th chain module C− k (G) as a finitely generated free :::::::::: F2-module with generators of the form: Ua1 1 · · · UaN N x where deg x − 2 i ai = k. Thus for each k, we can directly compute the k-th homology group H− k (G). (# of generators explodes as k decreases!)
  16. 20/34 Example (G = 31 , N = 5) k

    -6 -5 -4 -3 -2 -1 0 #{x} 2 10 27 40 30 10 1 rank C− k 622 360 192 90 35 10 1 rank H− k 1 0 1 0 1 0 1 Example (G = 61 , N = 8) k -11 ... -3 -2 -1 0 1 #{x} 1 ... 8,379 4,949 1,873 402 36 rank C− k 5,321,071 ... 24,659 8,165 2,161 402 36 rank H− k 0 ... 0 1 0 1 0
  17. 21/34 Proposition For any grid diagram G that represents a

    knot, H−(G) ∼ = F2[U1]. In particular, H− 0 (G) ∼ = F2. Thus the homology does not provide any information specific to the knot.
  18. 23/34 Bifiltration on C− ∗ (G) C− ∗ (G) admits

    a bifiltration. Namely, every monomial z = Ua1 1 · · · UaN N x ∈ C− ∗ (G), is assigned a bidegree (i, j) ∈ Z2 as i = −a1, the exponent of U1 in the coefficient of z, and j = AG (x) − a , where AG is the Alexander function. The first degree i is called the algebraic degree, and the second j is called the Alexander degree. It can be proved that ∂ is ::::::::::::: non-increasing for both i and j. There are several important concordance invariants such as τ, Vk and Υ that can be obtained from this bifiltration.
  19. 24/34 The invariant G0 (K) In year 2019, K. Sato

    introduced a concordance invariant G0(K) that unifies these invariants τ, Vk and Υ. We call a subset R ⊂ Z2 an LB region (LB = Left-Bottom) iff: If (i, j) ∈ R and (i ≤ i, j ≤ j) then (i , j ) ∈ R. For any LB region R, there is a corresponding subcomplex FRC− := Span F2 { z ∈ C− | (i(z), j(z)) ∈ R} where z is a monomial of the form z = Ua1 1 · · · UaN N x. Obviously ∂ is closed in FRC− from the definition of an LB region. For another LB region R ⊂ R, we have FR C− ⊂ FRC−. Thus C− ∗ (G) is filtered by the set of all LB regions.
  20. 26/34 For simplicity, we only describe the “shift-0 subset” G(0)

    0 (K) ⊂ G0(K). It is defined as the set of minimal LB regions, each containing a homological generator of H− 0 (G) ∼ = F2. Namely, R ∈ G(0) 0 (K) ⇔ ∃z ∈ FRC− 0 (G) s.t. 0 = [z] ∈ H− 0 (G), R is minimal w.r.t. the above property. If g is the genus of K, G0(K) = G(0) 0 (K) ∪ G(1) 0 (K) ∪ · · · ∪ G(g) 0 (K). Theorem (Sato ’19, Prop. 5.17) For any knot K, the invariant G0(K) determines all of τ, Vk and Υ.
  21. 27/34 i j τ i j k Vk Figure 3:

    Finding τ, Vk and Υ from G0 (K).
  22. 28/34 Computing G0 (K) From a joint work with Sato,

    we have developed an algorithm that computes G0(K) given any grid diagram as an input. Theorem (S.-Sato) There is an algorithm for computing the invariant G0(K). Corollary There is an algorithm for computing the invariants τ, Vk and Υ. As an application, we determined Υ for almost all knots of crossing number up to 11, including 39 knots whose Υ have been unknown. The missing ones are 10152, 11n31, 11n47, 11n77. For each of them, we have narrowed the candidates of Υ up to 3.
  23. 29/34 The idea for the computation The idea is same

    as in the computation of s(K). To see whether [z] ∈ FRC−, we mod out the generators that lies in R, and check whether the corresponding linear system has a solution. z z c ∂ sweep R R
  24. 30/34 The algorithm (sketch) Step 1. Compute one homological generator

    z ∈ C0(G). Step 2. Collect the candidates of G0(K) (it is assured to be finite). Step 3. Choose one maximal R. Take the basis of QRC0 = C0/FRC0 by modding out the generators of C0 that lie in R. Step 4. Compute the matrix A representing the differential dR : QRC−1 → QRC0. Step 5. Check whether Ax = vec(zR) has a solution. If it does, mark R as realizable. If it doesn’t, discard all regions that are included in R. Goto Step 3 if unchecked candidates exist. Step 6. Collect the realizable R’s that are minimal w.r.t. the inclusion ⊂, and we obtain G(0) 0 (K).
  25. 32/34 Future prospects There are many similarities seen between s

    and τ, though it is proved that they are not equal. Can we find some direct relations through computation? HFK is also extended over Z. We may find some more interesting phenomena by computing over Z. The program and the source code is planned to be published together with the paper (hopefully within this year!).
  26. 34/34 References I [1] M. Khovanov. “A categorification of the

    Jones polynomial”. In: Duke Math. J. 101.3 (2000), pp. 359–426. [2] E. S. Lee. “An endomorphism of the Khovanov invariant”. In: Adv. Math. 197.2 (2005), pp. 554–586. [3] J. Rasmussen. “Khovanov homology and the slice genus”. In: Invent. Math. 182.2 (2010), pp. 419–447. [4] C. Manolescu, P. Ozsv´ ath, and S. Sarkar. “A combinatorial description of knot Floer homology”. In: Ann. of Math. (2) 169.2 (2009), pp. 633–660. [5] C. Manolescu et al. “On combinatorial link Floer homology”. In: Geom. Topol. 11 (2007), pp. 2339–2412. [6] K. Sato. The ν+-equivalence classes of genus one knots. 2019. arXiv: 1907.09116 [math.GT].