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

Juhász, András; Marengon, Marco

doi:10.1007/s00029-017-0368-9

Cited by 15 publications

(14 citation statements)

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“…For knot Floer homology, cobordism maps are defined by Juhász [9] using contact geometry, and independently by Zemke [40] using elementary cobordisms, and together they [11] show that their definitions coincide. Juhász and Marengon [10] prove that the cobordism maps in [9] fit into a skein exact triangle, providing evidence that these cobordism maps are actually the maps in skein sequences. Thus, one approach to constructing the 2-morphisms mentioned above is to study the skein relations of tangle Floer homology further.…”

Section: Introductionmentioning

confidence: 93%

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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Section: Introductionmentioning

confidence: 93%

Skein relations for tangle Floer homology

Petkova

Wong

2020

Quantum Topol.

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“…was studied independently by Juhász and Marengon. In [19,Section 6], they also show that the isomorphism class of the resulting spectral sequence is a link type invariant.…”

Section: 4mentioning

confidence: 99%

On the functoriality of Khovanov–Floer theories

Baldwin

Hedden

Lobb

2019

Advances in Mathematics

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We introduce the notion of a Khovanov-Floer theory. Roughly, such a theory assigns a filtered chain complex over Z/2Z to a link diagram such that (1) the E 2 page of the resulting spectral sequence is naturally isomorphic to the Khovanov homology of the link; (2) this filtered complex behaves nicely under planar isotopy, disjoint union, and 1-handle addition; and (3) the spectral sequence collapses at the E 2 page for any diagram of the unlink. We prove that a Khovanov-Floer theory naturally yields a functor from the link cobordism category to the category of spectral sequences. In particular, every page (after E 1 ) of the spectral sequence accompanying a Khovanov-Floer theory is a link invariant, and an oriented cobordism in S 3 × [0, 1] between links in S 3 induces a map between each page of their spectral sequences, invariant up to smooth isotopy of the cobordism rel boundary.We then show that the spectral sequences relating Khovanov homology to Heegaard Floer homology and singular instanton knot homology are induced by Khovanov-Floer theories and are therefore functorial in the manner described above, as has been conjectured for some time. We further show that Szabó's geometric spectral sequence comes from a Khovanov-Floer theory, and is thus functorial as well. In addition, we illustrate how our framework can be used to give another proof that Lee's spectral sequence is functorial and that Rasmussen's invariant is a knot invariant. Finally, we use this machinery to define some potentially new knot invariants.

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“…In , Juhász and Marengon provide a partial computation of many elementary link cobordism maps from Juhász's sutured TQFT, which could be helpful in proving Conjecture . We will not pursue the conjecture in this paper.…”

Section: Introductionmentioning

confidence: 99%