Grid diagrams and Manolescu’s unoriented skein
exact triangle for knot Floer homology

Wong, C.-M. Michael

doi:10.2140/agt.2017.17.1283

Cited by 7 publications

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References 19 publications

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“…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.…”

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confidence: 79%

“…To better understand Manolescu's skein relation and the related conjectures, there are several approaches. One idea involves computing the maps in the skein relation combinatorially: The second author [38] gives a version of the skein sequence for grid homology, generalizing the results on quasi-alternating links [24,25] to Z-coefficients and giving a spectral sequence from a cube-of-resolutions complex with no diagonal maps. Lambert-Cole [21] exploits the computability in [38] to show that δ-graded knot Floer homology is invariant under Conway mutation by a large class of tangles.…”

Section: Introductionmentioning

confidence: 99%

“…Since tangle Floer homology shares some similarities with grid homology, our approach to proving Theorems 2 and 3 is similar to that in [38], and our approach to proving Theorem 5 is similar to that in [33,Chapter 9]. In particular, all maps involved are combinatorially computable.…”

Section: Introductionmentioning

confidence: 99%

“…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 ).…”

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confidence: 99%

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Skein relations for tangle Floer homology

Petkova

Wong

2020

Quantum Topol.

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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.

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confidence: 79%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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Skein relations for tangle Floer homology

Petkova

Wong

2020

Quantum Topol.

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Computing cobordism maps in link Floer homology and the reduced Khovanov TQFT

Juhász

Marengon

2017

Sel. Math. New Ser.

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We study the maps induced on link Floer homology by elementary decorated link cobordisms. We compute these for births, deaths, stabilizations, and destabilizations, and show that saddle cobordisms can be computed in terms of maps in a decorated skein exact triangle that extends the oriented skein exact triangle in knot Floer homology. In particular, we completely determine the Alexander and Maslov grading shifts.As a corollary, we compute the maps induced by elementary cobordisms between unlinks. We show that these give rise to a (1 + 1)-dimensional TQFT that coincides with the reduced Khovanov TQFT. Hence, when applied to the cube of resolutions of a marked link diagram, it gives the complex defining the reduced Khovanov homology of the knot. Finally, we define a spectral sequence from (reduced) Khovanov homology using these cobordism maps, and we prove that it is an invariant of the (marked) link.

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On Conway mutation and link homology

Lambert-Cole

2019

Advances in Mathematics

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We give a new, elementary proof that Khovanov homology with Z/2Z-coefficients is invariant under Conway mutation. This proof also gives a strategy to prove Baldwin and Levine's conjecture that δ-graded knot Floer homology is mutation-invariant. Using the Clifford module structure on HFK induced by basepoint maps, we carry out this strategy for mutations on a large class of tangles. Let L ′ be a link obtained from L by mutating the tangle T . Suppose some rational closure of T corresponding to the mutation is the unlink on any number of components. Then L and L ′ have isomorphic δ-graded HFK groups over Z/2Z as well as isomorphic Khovanov homology over Q. We apply these results to establish mutation-invariance for the infinite families of Kinoshita-Terasaka and Conway knots. Finally, we give sufficient conditions for a general Khovanov-Floer theory to be mutation-invariant.2010 Mathematics Subject Classification. 57M27; 57R58.The complex (CKh(D), d Kh ) possess two gradings, the quantum and homological grading, and the differential preserves the quantum grading and increases the homological grading by one. Thus the homology Kh(D) splits into the direct sum of bigraded modules Kh i,j (D) where i denotes the homological grading and j the quantum grading.Let p be a fixed basepoint in the plane contained in the diagram D. The basepoint p determines a chain map X p : CKh(D) → CKh(D) that squares to 0. The kernel of X p is a subcomplex and reduced Khovanov homology is its homology, with a shift in the quantum grading:It is an invariant of L up to isotopies supported away from p. While over Z the reduced homology depends on the component containing the basepoint, this is not true over Z/2Z.for every link L and any basepoint p. In particular, Kh(L) is well-defined independent of p.Consequently, when discussing reduced Khovanov homology over Z/2Z we will suppress any mention of the basepoint p.Khovanov homology satisfies Kunneth-type formulas for disjoint unions and connected sums. The following properties are well-known and the proofs are easy deductions from the definition of Khovanov homology and Proposition 2.1.Lemma 2.2. Let L 1 , L 2 be arbitrary links. Over Z/2Z there are isomorphismsfor any choice of connected sum.Khovanov homology satisfies an unoriented skein exact triangle. Fix a crossing of D and let D 0 , D 1 denote the 0-and 1-resolutions of D at this crossing. Since Khovanov homology is computed from a cube of resolutions, the complex CKh(D) is, up to a grading shift, the mapping cone of a chain map

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Grid diagrams and Manolescu’s unoriented skein exact triangle for knot Floer homology

Cited by 7 publications

References 19 publications

Skein relations for tangle Floer homology

Skein relations for tangle Floer homology

Computing cobordism maps in link Floer homology and the reduced Khovanov TQFT

On Conway mutation and link homology

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