Skein relations for tangle Floer homology

Abstract

In a previous paper, Vértesi and the first author used grid-like Heegaard diagrams to define tangle Floer homology, which associates to a tangle T a differential graded bimodule CT(T ). If L is obtained by gluing together T 1 , . . . , T m , then the knot Floer homology HFK(L) of L can be recovered from CT(T 1 ), . . . , CT(T m ). In the present paper, we prove combinatorially that tangle Floer homology satisfies unoriented and oriented skein relations, generalizing the skein exact triangles for knot Floer homology.2010 Mathematics Subject Classification. 57M58 (Primary); 57M25, 57M27 (Secondary).e 1 − 1 2 as δ-graded type DD structures. Analogous statements hold for type DA, AD, and AA structures.Remark. Following [33,34], our δ-gradings differ from those in [25,38] by a factor of −1.By taking the box tensor product, we immediately obtain a combinatorially computable unoriented skein exact triangle for knot Floer homology, recovering a version of the results in [24,38]. Suppose L ∞ , L 0 , and L 1 are three oriented links that are identical (after forgetting the orientations) except near a point, so that they form an unoriented skein triple. Let ℓ ∞ , ℓ 0 , and ℓ 1 be the number of components of L ∞ , L 0 , and L 1 respectively, and define neg(L k ), e 0 , and e 1 in a fashion analogous to neg(T k ), e 0 , and e 1 above.Corollary 4. For sufficiently large m, there exists a δ-graded exact trianglewhere V is a vector space of dimension 2 with grading 0, and W is a vector space of dimension 2 with grading −1.Remark. Due to a difference in the orientation convention, the arrows in the exact triangle point in the opposite direction from those in [24,25]. We follow the convention in [26][27][28]38], where the Heegaard surface is the oriented boundary of the α-handlebody.In another direction, Theorem 3 may also provide a way to further the development of knot Floer homology in the framework of categorification. Precisely, tangle Floer homology has been shown by Ellis, Vértesi, and the first author [5] to categorify the Reshetikhin-Turaev invariant for the quantum group U q (gl 1|1 ). This puts tangle Floer homology on a similar footing as the tangle formulation of Khovanov homology [3,13,39], which categorifies the Organization. We review the necessary algebraic background and the definition of tangle Floer homology in Section 2. We prove the ungraded unoriented skein relation, Theorem 2, in Section 3. We then determine the δ-gradings in Section 4 to prove the graded skein relation, Theorem 3. Theorems 5 and 6 is proven in Section 5.

On grid homology for lens space links: combinatorial invariance and integral coefficients

“…Suppose m = 3. Again, these cases are similar to those analyzed in the proof of Proposition 3.6, and are also similar to domains that were analyzed in [13]. Each such domain has a unique corner at which the domain has multiplicity one in three of the four regions around the corner, and multiplicity zero in the fourth.…”

“…. For the generator z of Figure 2.1 given by darkened circles, we can compute algebraically or via inspection of Figure 4.1 that C p,q (W (z)) = {(13, 0), (2, 1), (9, 2), (4, 3), (8,4), (0, 5), (10,6), (14,7), (6,8), (1,9), (5,10), (12,11), (7,12), (11,13), (3,14)}.…”

An unoriented skein relation via bordered–sutured Floer homology

“…In [MO08] (and in [Won17,PW20]), the exact triangles are equipped with an absolute δ-grading in Z. While we expect the grading in Corollary 1.2 to be related to this δ-grading when applied to Y = S 3 , we do not prove this in this paper.…”

“…There are currently several theories related to Heegaard Floer homology that provide a suitable gluing theorem; in this paper, we focus on one such theory, bordered-sutured Floer homology, defined by Zarev [Zar11]. Petkova and the second author [PW20] have proven a similar result for tangle Floer homology, a combinatorial tangle invariant defined by Petkova and Vértesi [PV16], similar to grid homology, for tangles in S 2 × [0, 1]; meanwhile, Zibrowius [Zib17] has given an alternative proof of Manolescu's result for links in S 3 using peculiar modules, which are invariants of tangles in B 3 . The exact triangle in the present paper applies to tangles in any 3-manifold.…”