On the decategorification of Ozsváth and Szabó's bordered theory for knot Floer homology

Manion, Andrew

doi:10.4171/qt/123

Cited by 15 publications

(13 citation statements)

References 26 publications

(51 reference statements)

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“…Theorem 1.2 generalizes the path-algebra descriptions of B(n, k, S) for n = 1, 2 given in [OSz18,Section 3.5]. Theorems 1.1 and 1.2 have already seen use in [Man19,Man17], as well as in [AD18,Section 4.1]. Theorem 1.2 will be especially useful in [MMW19], where we use it to define a quasi-isomorphism from B(n, k, S) to a certain generalized strands algebra A(n, k, S) as discussed in the motivational section below.…”

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confidence: 64%

“…Remark 3.17. In [Man19], the algebras B r (n, k, S) and B l (n, k, S) were referred to as C r (n, k, S) and C l (n, k, S), following old notation of Ozsváth-Szabó, and shown to categorify tensor products V ⊗S of the vector representation V of U q (gl(1|1)) and its dual (depending on the orientations S). The terminology in the next definitions is also due to Ozsváth-Szabó, although it does not appear explicitly in [OSz18].…”

Section: Idempotent-truncated Algebrasmentioning

confidence: 99%

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Generators, Relations, and Homology for Ozsváth–szabó’s Kauffman-States Algebras

Manion¹,

Marengon²,

Willis³

2020

Nagoya Math. J.

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We give a generators-and-relations description of differential graded algebras recently introduced by Ozsváth and Szabó for the computation of knot Floer homology. We also compute the homology of these algebras and determine when they are formal. IntroductionHeegaard Floer homology [OSz04c, OSz04b] is a powerful family of invariants for 3-and 4-manifolds. It originated from the study of Seiberg-Witten theory and Donaldson theory, although its methods involve holomorphic curves rather than gauge theory, and it shares these theories' applicability to the exotic world of smooth 4-manifolds. Compared with its gauge-theoretic relatives, Heegaard Floer homology is often the easiest for computations, and many forms of Heegaard Floer homology have now been given combinatorial definitions.One form of Heegaard Floer homology, called knot Floer homology (or HF K), assigns graded abelian groups to knots and links in 3-manifolds [OSz04a, Ras03]. Like Khovanov homology [Kho00], HF K is especially well-adapted to the study of problems in knot theory with a 4-dimensional character, such as the structure of the knot concordance group. There are many interesting similarities between HF K and Khovanov homology; for example, while the Euler characteristic of Khovanov homology is the Jones polynomial, the Euler characteristic of HF K is the Alexander polynomial.Combined with constructions of the Jones and Alexander polynomial from the representation theory of U q (sl(2)) and U q (gl(1|1)) respectively, this analogy suggests a close link between Heegaard Floer homology and categorifications of the Witten-Reshetikhin-Turaev topological quantum field theory (TQFT) invariants, see e.g. [Ras05,DGR06]. Indeed, both Donaldson-Floer theories in 4 dimensions and Witten-Reshetikhin-Turaev theories in 3 dimensions were initial motivations for the mathematical study of TQFTs, and Heegaard Floer homology offers a promising framework for understanding the relationship between these two types of theories.Among Heegaard Floer theories, HF K admits an especially wide variety of combinatorial descriptions, some allowing very fast computations. In particular, Ozsváth-Szabó have a computer program [OSzb] that can compete with Bar-Natan's fast Khovanov homology program [BN07]. Ozsváth-Szabó's program can quickly compute HF K for most knots with up to around 40 or 50 crossings, and can even handle the larger 90+ crossing examples from the paper [FGMW10].Ozsváth-Szabó's program is based on an exciting new description of HF K [OSz18, OSz17, OSza, OSzc] in the algebraic language of bordered Floer homology, an extended TQFT approach to Heegaard Floer homology.

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confidence: 64%

Section: Idempotent-truncated Algebrasmentioning

confidence: 99%

Generators, Relations, and Homology for Ozsváth–szabó’s Kauffman-States Algebras

Manion¹,

Marengon²,

Willis³

2020

Nagoya Math. J.

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“…It is natural to ask, then, whether Ozsváth-Szabó's theory in [OS16] for knot Floer homology can be viewed as categorifying some U q (gl(1|1))-representation setup. This is indeed the case and will be addressed in [Man16].…”

Section: Introductionmentioning

confidence: 76%

Khovanov–Seidel quiver algebras and Ozsváth–Szabó's bordered theory

Manion

2017

Journal of Algebra

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We investigate a relationship between Ozsváth and Szabó's bordered theory [OS16] and the algebras and bimodules constructed by Khovanov and Seidel in [KS02]. Specifically we show that (a variant of) a special case of Ozsváth-Szabó's algebras has a quotient which is isomorphic to the Khovanov-Seidel quiver algebra with coefficients in Z/2Z. Furthermore, we show that after induction and restriction of scalars, the dg bimodule over quiver algebras associated to a crossing by Khovanov-Seidel is homotopy equivalent to Ozsváth-Szabó's DA bimodule for the crossing in this special case. Khovanov-Seidel's constructionWe briefly review some definitions from [KS02].

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“…It is interesting to compare the ideas described in this paper to the combinatorial tangle Floer theory by Petkova and Vértesi and the algebraic tangle homology theory by Ozsváth and Szabó as well as their corresponding decategorifications in terms of the representation theory of

{scriptU}_{q} false(frakturgl (1 | 1) false)

. In fact, the definition of our generalised peculiar modules

{CFT}^{-}

is primarily inspired by the invariants from , see Remark .…”

Section: Introductionmentioning

confidence: 99%

Peculiar modules for 4‐ended tangles

Zibrowius

2020

Journal of Topology

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With a 4‐ended tangle T, we associate a Heegaard Floer invariant prefixCFT∂false(Tfalse), the peculiar module of T. Based on Zarev's bordered sutured Heegaard Floer theory (Zarev, PhD Thesis, Columbia University, 2011), we prove a glueing formula for this invariant which recovers link Floer homology HFL̂. Moreover, we classify peculiar modules in terms of immersed curves on the 4‐punctured sphere. In fact, based on an algorithm of Hanselman, Rasmussen and Watson (Preprint, 2016, arXiv:1604.03466v2), we prove general classification results for the category of curved complexes over a marked surface with arc system. This allows us to reinterpret the glueing formula for peculiar modules in terms of Lagrangian intersection Floer theory on the 4‐punctured sphere. We then study some applications: Firstly, we show that peculiar modules detect rational tangles. Secondly, we give short proofs of various skein exact triangles. Thirdly, we compute the peculiar modules of the 2‐stranded pretzel tangles T2n,−false(2m+1false) for n,m>0 using nice diagrams. We then observe that these peculiar modules enjoy certain symmetries which imply that mutation of the tangles T2n,−false(2m+1false) preserves δ‐graded, and for some orientations even bigraded link Floer homology.

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On the decategorification of Ozsváth and Szabó's bordered theory for knot Floer homology

Cited by 15 publications

References 26 publications

Generators, Relations, and Homology for Ozsváth–szabó’s Kauffman-States Algebras

Generators, Relations, and Homology for Ozsváth–szabó’s Kauffman-States Algebras

Khovanov–Seidel quiver algebras and Ozsváth–Szabó's bordered theory

Peculiar modules for 4‐ended tangles

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