Khovanov homology and knot Floer homology for pointed links

Baldwin, John A.; Levine, Adam Simon; Sarkar, Santanu

doi:10.1142/s0218216517400041

Cited by 23 publications

(34 citation statements)

References 34 publications

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“…Dunfield-Gukov-Rasmussen [12] have conjectured the existence of a corresponding specialization spectral sequence from (uncolored) HOMFLY-PT homology to knot Floer homology [41,44]. Recent developments related to this conjecture and the present paper include work by Dowlin [11] and Baldwin-Levine-Sarkar [1], who match up structural properties of both types of link homologies, and Ellis-Petkova-Vértesi [13], whose results can be interpreted as saying that knot Floer homology (and its extension to tangles) is already a gl 1|1 homology.…”

Section: Introductionmentioning

confidence: 81%

Exponential growth of colored HOMFLY-PT homology

Wedrich

2019

Advances in Mathematics

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We define reduced colored sl N link homologies and use deformation spectral sequences to characterize their dependence on color and rank. We then define reduced colored HOMFLY-PT homologies and prove that they arise as large N limits of sl N homologies. Together, these results allow proofs of many aspects of the physically conjectured structure of the family of type A link homologies. In particular, we verify a conjecture of Gorsky, Gukov and Stošić about the growth of colored HOMFLY-PT homologies. arXiv:1602.02769v1 [math.GT] 8 Feb 2016 Theorem 2. . Let K be a labelled knot. For sufficiently large N , there are isomorphisms i+2N j=I h−j=Jbetween a grading-collapsed version of reduced colored HOMFLY-PT homology and reduced colored sl N homology. See Theorem 3.46.Theorem 2 relies on the fact that the reduced colored HOMFLY-PT homology is finite-dimensional for knots, see Proposition 3.45. On the contrary, the original unreduced colored HOMFLY-PT homology of Mackaay-Stošić-Vaz [36] and Webster-Williamson [53] is always infinite-dimensional. Nevertheless, we find an analogue of Theorem 1, which relates it to unreduced colored sl N homologies and -more generally -their Σ-deformed versions, which have been defined by Wu in [58] and further studied by Rose and the author in [49].Theorem 3. Let L be a labelled link. There is a spectral sequencefrom unreduced colored HOMFLY-PT homology to unreduced, Σ-deformed colored sl N homology. See Theorem 3.22.1.2. The physical structure. The existence of reduced colored sl N and HOMFLY-PT homologies and the theorems describing their relationships, although new, will not come as a surprise to the expert. In fact, they are at the very centre of current interest in the study of link homologies from the prespective of theoretical physics. Since Gukov-Schwarz-Vafa's physical interpretation of Khovanov-Rozansky homology as a space of BPS states [22], there has been significant cross-fertilization between the mathematical and physical sides of this field. A particularly interesting outcome is a package of conjectures about the structure of the family of colored sl N and HOMFLY-PT homologies, which has been developed over a decade in a series of papers by Dunfield-Gukov-Rasmussen [12], Gukov-Walcher [24], Gukov-Stošić [23], Gorsky-Gukov-Stošić [18] and most recently Gukov-Nawata-Saberi-Stošić-Su lkowski [20]. Thanks to recent advances in understanding the higher representation theory underlying link homologies (in particular categorical skew Howe duality), which led to the convergence of several different approaches to link homologies [7,42,49,37], we are now in a position to prove a significant part of the physically conjectured structure. In the following theorems, K k denotes a knot labelled with the k th exterior power of the vector representation.Theorem 4. There is a spectral sequencewhich preserves the homological grading. See Corollary 3.47.This implies that the colored HOMFLY-PT homologies of a knot grow at least exponentially in color, as predicted in [18]. A related result has ...

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Section: Introductionmentioning

confidence: 81%

Exponential growth of colored HOMFLY-PT homology

Wedrich

2019

Advances in Mathematics

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“…While the discussion above seems to suggest that there may be a spectral sequence relating Kh(L) and HFK(L) that comes from iterating Manolescu's skein relation, Baldwin and Levine [1] discover that the E 2 page of the spectral sequence they so construct is not even a link invariant. However, one may be able to relate the two theories with some modifications: Baldwin, Levine, and Sarkar [2] construct another spectral sequence that converges to HFK(L) ⊗ V n for some module V of rank 2 and some integer n, where the differential D 0 counts some of the holomorphic polygons in Manolescu's unoriented skein sequence. They conjecture that the E 1 page of this spectral sequence coincides with a variant of Khovanov homology for pointed links, the proof of which would imply a version of the following conjecture, first formulated by Rasmussen [35] for knots: Conjecture 1.…”

Section: Introductionmentioning

confidence: 99%

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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“…This spectral sequence is the framed instanton theory analogue of the spectral sequence in [10]. They moreover conjecture a relation between Hd(D, ω) and a twisted Khovanov homology similar to those in [2], [6], and [14], which is an invariant of links with marking data, and which also has a spectral sequence relating it to the framed instanton homology.…”

Section: Figurementioning

confidence: 74%

Khovanov homology and binary dihedral representations for marked links

Gong

2019

Journal of Geometry and Physics

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We introduce a version of Khovanov homology for alternating links with marking data, ω, inspired by instanton theory. We show that the analogue of the spectral sequence from Khovanov homology to singular instanton homology introduced in [10] for this marked Khovanov homology collapses on the E 2 page for alternating links. We moreover show that for non-split links the Khovanov homology we introduce for alternating links does not depend on ω; thus, the instanton homology also does not depend on ω for non-split alternating links.Finally, we study a version of binary dihedral representations for links with markings, and show that for links of non-zero determinant, this also does not depend on ω.

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Khovanov homology and knot Floer homology for pointed links

Cited by 23 publications

References 34 publications

Exponential growth of colored HOMFLY-PT homology

Exponential growth of colored HOMFLY-PT homology

Skein relations for tangle Floer homology

Khovanov homology and binary dihedral representations for marked links

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