Divisibility of Lee’s class and its relation with Rasmussen’s invariant

Tsuda-Gakugei Topology Workshop
https://edu.tsuda.ac.jp/~kunotti/tgtw2018.pdf

Transcript

1. Divisibility of Lee’s class and its relation with Rasmussen’s invariant 1 Taketo Sano The University of Tokyo 2019-09-28 Tsuda-Gakugei Topology Workshop 1https://arxiv.org/abs/1812.10258
2. Contents 1. Introduction 2. Khovanov homology 3. Lee’s classes in Hh,t 4. The invariant sc(K) 5. Coincidence with s 6. Future prospects
3. Contents 1. Introduction 2. Khovanov homology 3. Lee’s classes in Hh,t 4. The invariant sc(K) 5. Coincidence with s 6. Future prospects
4. Overview
5. Khovanov homology Khovanov homology HKh is a bigraded link homology theory, constructed combinatorially from a planar link diagram. For a link L, the graded Euler characteristic of H·· Kh (L; Q) gives the (unnormalized) Jones polynomial of L. HKh(L; Q) L V (L) graded Euler characteristic Jones Polynomial Khovanov homology
6. Figure 1: HKh (K; Q) for K = 01, 31, 41
7. Figure 2: HKh (K; Q) for K = 942
8. Lee homology Lee homology HLee is a variant of Khovanov homology. For a knot diagram D, there are two distinct classes [α], [β] constructed combinatorially from D, and they form a basis of HLee(D; Q). Theorem ([Lee05, Theorem 4.2]) HLee(D; Q) = [α], [β] ∼ = Q2
9. Rasmussen’s s-invariant Rasmussen introduced the s-invariant based on Q-Lee homology. Theorem ([Ras10]) 1. s defines a homomorphism from the knot concordance group in S3 to 2Z: s : Conc(S3) → 2Z. 2. s gives a lower bound of the slice genus: |s(K)| ≤ 2g∗(K). 3. If K is a positive knot, then s(K) = 2g∗(K) = 2g(K).
10. With the above three properties of s, one obtains: Corollary (The Milnor Conjecture, [Mil68]) The (smooth) slice genus and the unknotting number of the (p, q)-torus knot are both equal to (p − 1)(q − 1)/2. Remark The Milnor Conjecture was first proved by Kronheimer and Mrowka in [KM93] using gauge theory, but Rasmussen’s result was notable since it provided a purely combinatorial proof.
11. Our observation Lee homology and the two classes [α], [β] can be similarly defined over Z. Do they still form a basis of HLee(D; Z)? We created a computer program that calculates HLee(D; Z), and the components of [α], [β] with respect to some computed basis.
12. Figure 3: Computational results 2-powers seem to appear in the components of [α], [β]. Thus they do not form a basis over Z.
13. Question Where does these 2-powers come from? Can we extract some information that were hidden when considered over Q?
14. Our results We define the 2-divisibility k2(D) of [α] (modulo torsions) by the exponent of its 2-power factor. By inspecting the variance of k2(D) under the Reidemeister moves, we prove the following: Theorem (S.) s2(K) := 2k2(D) + w(D) − r(D) + 1 is a knot invariant, where w is the writhe, and r is the number of Seifert circles. Again by computational experiments, we confirmed that s2 = s holds for all prime knots of crossing number up to 11. Are the two invariants equal?
15. The following theorem support the affirmative answer. Theorem (S.) s2 possesses properties common to s, namely: 1. s2 is a knot concordance invariant in S3. 2. For any knot K, |s2(K)| ≤ 2g∗(K). 3. If K is a positive knot, then s2(K) = 2g∗(K) = 2g(K). In particular, the Milnor conjecture follows as a corollary.
16. We consider the problem in a more generalized setting. Khovanov’s original theory and other variant theories can be unified by a one parameter family of Khovanov-type homology theories {Hc(−; R)}c∈R over an arbitrary commutative ring R. HKh = Hc=0, HLee = Hc=2. For each c ∈ R, Lee’s classes [α], [β] of a knot diagram D can also be defined in Hc(D; R). Assuming that R is an integral domain and c is ::::::::: non-zero, :::::::::::: non-invertible, we define the c-divisibility kc(D; R) of [α] (modulo torsions) by the exponent of its c-power factor. Then again sc(K; R) := 2kc(D; R) + w(D) − r(D) + 1 is a knot invariant, and the previous results hold.
17. Especially for (R, c) = (Q[h], h) we have the following: Theorem (S.) sh(−; Q[h]) = s. Remaining Question Are all sc(−; R) equal to s? If so, then s is given a description by the combination of the divisibility of Lee’s class and some classical diagram invariants. If not, then we obtain a (potentially) new invariant.
18. Conventions In this talk, all knots and links are assumed to be oriented. For simplicity, we mainly focus on knots, but many of the results can be generalized to links.
19. Contents 1. Introduction 2. Khovanov homology 3. Lee’s classes in Hh,t 4. The invariant sc(K) 5. Coincidence with s 6. Future prospects
20. Frobenius algebra Let R be a commutative ring with unity. A Frobenius algebra over R is a quintuple (A, m, ι, ∆, ε) satisfying: 1. (A, m, ι) is an associative R-algebra with multiplication m : A ⊗ A → A and unit ι : R → A, 2. (A, ∆, ε) is a coassociative R-coalgebra with comultiplication ∆ : A → A ⊗ A and counit ε : A → R, and 3. the Frobenius relation holds: ∆ ◦ m = (id ⊗ m) ◦ (∆ ⊗ id) = (m ⊗ id) ◦ (id ⊗ ∆).
21. A commutative Frobenius algebra A gives a 1+1 TQFT FA : Cob2 −→ ModR, by mapping: Objects: · · · r −→ A ⊗ · · · ⊗ A r Morphisms:
22. Construction of the chain complex Let D be a link diagram with n crossings. The 2n resolutions of the crossings yields a commutative cubic diagram in Cob2. By applying FA we obtain a commutative cubic diagram in ModR. Then we turn this cube skew commutative by appropriately adjusting the signs of the edge maps. Finally by folding the cube into a sequence we obtain the chain complex CA(D) and its homology HA(D).
23. Khovanov homology and its variants Khovanov’s original theory is given by A = R[X]/(X2). Other variant theories are given by: A = R[X]/(X2 − 1) → Lee’s theory A = R[X]/(X2 − hX) → Bar-Natan’s theory Khovanov unified these theories in [Kho06] by considering the following special Frobenius algebra with h, t ∈ R: Ah,t = R[X]/(X2 − hX − t). Denote the corresponding chain complex by Ch,t(D; R) and its homology by Hh,t(D; R). It is proved that the isomorphism class of Hh,t(D; R) is a link invariant.
24. Contents 1. Introduction 2. Khovanov homology 3. Lee’s classes in Hh,t 4. The invariant sc(K) 5. Coincidence with s 6. Future prospects
25. Factoring the quadratic polynomial In order to generalize Lee’s classes [α], [β] as elements in Hh,t(D; R), we assume (R, h, t) satisfies the following condition: Condition X2 − hX − t factors into linear polynomials in R[X]. ⇔ There exists c ∈ R such that h2 + 4t = c2 and (h ± c)/2 ∈ R. Assuming that this condition holds, fix one square root c = √ h2 + 4t. Let X2 − hX − t = (X − u)(X − v) with c = v − u, and define a = X − u, b = X − v ∈ A.
26. Regrading A as a free module over R with basis {1, X}, the multiplication and comultiplication are given by: m(X ⊗ X) = hX + t1, ∆(X) = X ⊗ X + t1 ⊗ 1, m(X ⊗ 1) = X, ∆(1) = X ⊗ 1 + 1 ⊗ X − h1 ⊗ 1 m(1 ⊗ X) = X m(1 ⊗ 1) = 1 For a and b, the above operations diagonalize as: m(a ⊗ a) = ca, ∆(a) = a ⊗ a, m(a ⊗ b) = 0, ∆(b) = b ⊗ b m(b ⊗ a) = 0 m(b ⊗ b) = −cb
27. Definition of Lee’s classes The cycle α ∈ Ch,t(D; R) is defined by the following procedure: 1. Resolve all crossings of D in the orientation preserving way. 2. Color the regions of R2 \ {circles} in the checkerboard fashion. 3. To each circle, assign a if it sees a black region to the left, otherwise assign b. 4. Define α as the corresponding tensor product of a’s and b’s in Ch,t(D; R). The cycle β ∈ Ch,t(D; R) is defined similarly with the orientation of D reversed.
28. Reduction of parameters Proposition For another (h , t ) such that √ h 2 + 4t = c, the corresponding group Hh ,t (D; R) is naturally isomorphic to Hh,t(D; R). Under the isomorphism the Lee classes correspond one-to-one. Thus we denote the isomorphism class of Hh,t(D; R), h2 + 4t = c by Hc(D; R), and regard [α], [β] as elements in Hc(D; R). Remark HKh = Hc=0, HLee = Hc=2.
29. Generalizing Lee’s theorem Proposition If : c :: is ::::::::: invertible in R, then Hc(D; R) = [α], [β] . Proof. If c is invertible, then we may take {a, b} as a basis of A = 1, X , since the transformation matrix −u −v 1 1 has determinant v − u = c. The rest follows by applying Wehrli’s proof given in [Weh08]. Remark Lee’s theorem HLee(D; Q) = [α], [β] is the special case (R, c) = (Q, 2). The original proof cannot be applied directly, since it uses Hodge theory and requires that R is a field.
30. Behavior under Reidemeister moves To each Reidemeister move D → D , an isomorphism is given explicitly ρ : Hh,t(D; R) → Hh,t(D ; R). It can be shown that ρ induces an isomorphism ρ : Hc(D; R) → Hc(D ; R). Rasmussen proved that, in Q-Lee theory, Lee’s classes correspond one-to-one (up to unit) under the above isomorphisms. The following proposition is a generalization of this fact. Proposition Suppose D, D are two diagrams related by a single Reidemeister move. Let ρ be the corresponding isomorphism. (...)
31. Proposition (continued) There exists some j ∈ {0, ±1} and ε, ε ∈ {±1} such that Lee’s classes of D and D are related as: [α ] = εcj · ρ[α], [β ] = ε cj · ρ[β]. Moreover the exponent j is given by j = ∆r − ∆w 2 where r denotes the number of Seifert circles, w denotes the writhe, and the prefixed ∆ is the difference of the corresponding values for D and D . Proof. Check all possible patterns of [α], [β] and those images under ρ.
32. Summary We defined an one-parameter family of Khovanov-type homology theories {Hc(−; R)}c∈R. Given a knot diagram D, Lee’s classes [α], [β] ∈ Hc(D; R) can be defined for each c ∈ R. If (and only if) : c ::: is ::::::::: invertible, Lee’s classes [α], [β] form a basis of Hc(D; R) and they are invariant (up to unit) under the Reidemeister moves. Thus the situation is completely analogous to Q-Lee theory when c is invertible. As in the case of Z-Lee theory, our concern is when : c : is :::: not ::::::::: invertible. Remark All results in this section can be generalized to link diagrams.
33. Contents 1. Introduction 2. Khovanov homology 3. Lee’s classes in Hh,t 4. The invariant sc(K) 5. Coincidence with s 6. Future prospects
34. An assumption and some notations Assumption For the remainder of this slide, we assume R is an integral domain, and c ∈ R is non-zero and non-invertible. Notation We denote the quotient of Hc(D; R) by it torsion submodule by Hc(D; R)f := Hc(D; R)/Tor. By abuse of notation, we denote the images of [α], [β] in Hc(D; R)f by the same symbols.
35. Recall: Computational results For (R, c) = (Z, 2), the classes [α], [β] seemed to be divisible by 2-powers. For a general (R, c), we may expect that [α], [β] are divisible by c-powers.
36. The c-divisibility of Lee’s class Definition Let M be an R-module, and z ∈ M. Define the c-divisibility kc(z) of z by: kc(z) := max{ k ≥ 0 | z ∈ ckM }. Remark There is a filtration of M given by: M ⊃ c · M ⊃ · · · ⊃ ck · M ⊃ · · · kc(z) gives the maximal filtration level that contains z.
37. Definition For any knot diagram D, we define kc(D) = kc([α]) where [α] ∈ Hc(D; R)f . Remark It can be shown that kc([α]) = kc([β]). Proposition If D is positive, then kc(D) = 0.
38. Behavior of kc under the Reidemeister moves Proposition Let D, D be two diagrams of the same knot. Then ∆kc = ∆r − ∆w 2 , where the prefixed ∆ is the difference of the corresponding numbers for D and D . Proof. In the previous section, we had [α ] = ±cj · ρ[α], j = ∆r − ∆w 2 so kc([α ]) = j + kc(ρ[α]) = j + kc([α]).
39. The invariant sc (K) Corollary The number kc(D) − r(D) − w(D) 2 is an knot invariant. Definition For a knot K, define sc(K) := 2kc(D) − r(D) + w(D) + 1
40. Proposition ( ) If K is positive, then sc(K) = 2g(S), where S is a Seifert surface of K obtained by applying the Seifert’s algorithm to a positive diagram of K. Proof. Let D be a positive diagram of K. Since D is positive, kc(D) = 0, w(D) = n(D). On the other hand, we have χ(S) = 1 − 2g(S) = r(D) − n(D), so sc(K) = −r(D) + n(D) + 1 = 2g(S).
41. Behavior under cobordisms Proposition If S is an oriented connected cobordism between knots K, K , then |sc(K ) − sc(K)| ≤ −χ(S). Proof. Decompose S into elementary cobordisms such that each factor corresponds to a Reidemeister move or a Morse move. Inspect the successive images of [α] at each level.
42. Theorem (S.) 1. sc is a knot concordance invariant in S3. 2. For any knot K, |sc(K)| ≤ 2g∗(K). 3. If K is a positive knot, then sc(K) = 2g∗(K) = 2g(K). Proof. 1. If K, K are concordant, then by definition they are cobordant by an annulus S. Obviously χ(S) = 0. 2. Let S be a g∗(K) realizing surface with a small disk removed inside. Then −χ(S) = 2g∗(K). 3. Let S be a Seifert surface of K as in Prop ( ). Then sc(K) ≤ 2g∗(K) ≤ 2g(K) ≤ 2g(S) = sc(K).
43. Corollary (The Milnor conjecture) The slice genus and the unknotting number of the (p, q)-torus knot Tp,q are both equal to (p − 1)(q − 1)/2. Proof. With the positive braid representation of Tp,q, we have r(D) = p, w(D) = n(D) = (p − 1)q so g∗(Tp,q) = sc(Tp,q) = (p − 1)(q − 1)/2. For the unknotting number, g∗(K) ≤ u(K) holds in general, and we can show that Tp,q can be unknotted by (p − 1)(q − 1)/2 crossing changes.
44. Remark The definition of kc(D) and sc(K) can be extended to links. In particular |sc(L ) − sc(L)| ≤ −χ(S) holds, and it follows that sc(L) is a link concordance invariant.
45. Contents 1. Introduction 2. Khovanov homology 3. Lee’s classes in Hh,t 4. The invariant sc(K) 5. Coincidence with s 6. Future prospects
46. The canonical generator [ζ] for (R, c) = (Q[h], h) Now we focus on the case (R, c) = (Q[h], h), deg h = −2 and prove that sh(−; Q[h]) coincides with s. Recall that in general [α], [β] do not form a basis of Hc(D; R)f . In the above case, we can “normalize” them to obtain a class [ζ] such that {[ζ], X[ζ]} is a basis of Hc(D; R)f . Remark X denotes an action on Hc (D; R) defined by merging a circled labeled X to a neighborhood of a fixed point of D.
47. Proposition There is a unique class [ζ(D)] ∈ Hh(D; R)f such that [ζ], X[ζ] form a basis of Hh(D; R)f , and are invariant under the Reidemeister moves. [α], [β] can be described as [α] = hk( (h/2)[ζ] + X[ζ]) [β] = (−h)k(−(h/2)[ζ] + X[ζ]), where k = kh(D). Remark Unlike [α] and [β], the definition of [ζ] is non-constructive. Note that the h-divisibility of [α], [β] can be seen explicitly.
48. The homomorphism property of sh Using the class [ζ], we can prove: Proposition 1. sh(K) = −sh(K∗) 2. sh(K#K ) = sh(K) + sh(K ). Thus the invariant sh(−; Q[h]) defines a homomorphism sh : Conc(S3) → 2Z. Remark By adjusting the signs of the isomorphism ρ and the cobordism map, we can show that the class [ζ] itself is a knot concordance invariant.
49. Coincidence with the s-invariant Theorem (S.) sh(−; Q[h]) coincides with Rasmussen’s s. Proof. Both s and sh changes sign by mirroring, so it suffices to prove s(K) ≥ sh(K). Recall that s(K) is defined by the (filtered) q-degree of HLee(D; Q). On the other hand, q-degree on Hh(D; Q[h]) gives a strict grading. There is a q-degree non-decreasing map π : Hh(D; Q[h]) → HLee(D; Q) induced from Q[h] → Q, h → 2. (...)
50. Proof continued. Denote by [α2], [αh] the α-classes of D in HLee(D; Q), Hh(D; Q[h]) respectively. Then by definition π[αh] = [α2]. Let [αh] = hk[αh ] with k = kh(D). Since deg h = −2, qdegh ([αh ]) = qdegh ([αh]) + 2k = w(D) − r(D) + 2k, and we have s(K) = qdeg([α2]) + 1 = qdeg(π[αh]) + 1 = qdeg(π[αh ]) + 1 ≥ qdegh ([αh ]) + 1 = w(D) − r(D) + 2k + 1 = sh(K).
51. Corollary s(K) = qdegh [ζ(K)] − 1. Remark The construction of [ζ(K)] and the homomorphism property of sc also holds for (R, c) = (Z, 2) or (R, c) = (F[h], h) where F is an arbitrary field of char F = 2. Remark The results of this section has only been proved for knots.
52. Contents 1. Introduction 2. Khovanov homology 3. Lee’s classes in Hh,t 4. The invariant sc(K) 5. Coincidence with s 6. Future prospects
53. Question (1) Are all sc equal to s? Remark (1) The s-invariant can be defined over any field F of ::::::::: char F = 2. We can prove that s(−; F) = sh (−; F[h]). It is an open question whether s(−; F) for char F = 2 are all equal or not [LS14, Question 6.1]. If [Question 1] is solved affirmatively, then it follows that s(−; F) are all equal. Remark (2) In [LS14], an alternative definition of s over any field F (including char F = 2) is given, based on the :::::: filtered ::::::::: Bar-Natan ::::::::: homology. C.Seed showed by direct computation that K = K14n19265 has s(K; Q) = 0 but s(K; F2 ) = −2.
54. Question (2) Can we construct [ζ] ∈ Hc(D; R) for any (R, c)? The existence of [ζ] ∈ Hh(D; Q[h]) was the key to prove s = sh. If such class exists in general, then we can expect that Question (1) can also be solved. However the current proof for (R, c) = (Q[h], h) cannot be applied to the general case. Maybe we can find a more geometric (or combinatorial) construction.
55. Question (3) Does s = sc(−; Q[h]) also hold for links? The definition of s for links is given by Beliakova and Wehrli in [BW08]. Our sc can also be defined for links, so the question makes sense. Maybe we can construct the canonical generators [ζ1], · · · , [ζ2 ] of Hh(D; Q[h]) such that the α-classes {[α(D, o)]}o can be described by them.
56. Thank you! https://arxiv.org/abs/1812.10258
57. References I [BW08] A. Beliakova and S. Wehrli. “Categorification of the colored Jones polynomial and Rasmussen invariant of links”. In: Canad. J. Math. 60.6 (2008), pp. 1240–1266. [Kho06] M. Khovanov. “Link homology and Frobenius extensions”. In: Fund. Math. 190 (2006), pp. 179–190. [KM93] P. B. Kronheimer and T. S. Mrowka. “Gauge theory for embedded surfaces. I”. In: Topology 32.4 (1993), pp. 773–826. [Lee05] E. S. Lee. “An endomorphism of the Khovanov invariant”. In: Adv. Math. 197.2 (2005), pp. 554–586. [LS14] R. Lipshitz and S. Sarkar. “A refinement of Rasmussen’s S-invariant”. In: Duke Math. J. 163.5 (2014), pp. 923–952. [Mil68] J. Milnor. Singular points of complex hypersurfaces. Annals of Mathematics Studies, No. 61. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968, pp. iii+122.
58. References II [Ras10] J. Rasmussen. “Khovanov homology and the slice genus”. In: Invent. Math. 182.2 (2010), pp. 419–447. [Weh08] S. Wehrli. “A spanning tree model for Khovanov homology”. In: J. Knot Theory Ramifications 17.12 (2008), pp. 1561–1574.