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

Manion, Andrew

doi:10.1016/j.jalgebra.2017.05.029

Cited by 12 publications

(11 citation 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: 62%

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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: 62%

“…Remark 3.11. In [Man17], the meaning of "refined" and "unrefined" grading was reversed. Here we use terminology following [LOT18, Section 3.3] in line with [MMW19] where we relate the gradings on Ozsváth-Szabó's algebras with group-valued gradings on strands algebras.…”

Section: Alexander and Maslov Gradingsmentioning

confidence: 99%

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

Manion¹,

Marengon²,

Willis³

2020

Nagoya Math. J.

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“…Gradings. The algebra B(n, k) has a grading by 1 2 Z n that we call the refined Alexander multi-grading, as well as a grading by Z 2n considered in [Man17,MMW19a,MMW19b] and called the unrefined Alexander multigrading. Our terminology here contrasts with that of [Man17] but is more in line with the use of "refined" and "unrefined" in [LOT18].…”

Section: Basismentioning

confidence: 99%

“…The algebra B(n, k) has a grading by 1 2 Z n that we call the refined Alexander multi-grading, as well as a grading by Z 2n considered in [Man17,MMW19a,MMW19b] and called the unrefined Alexander multigrading. Our terminology here contrasts with that of [Man17] but is more in line with the use of "refined" and "unrefined" in [LOT18]. Indeed, it is shown in [MMW19b] that B(n, k) is quasi-isomorphic to a generalized bordered strands algebra A(n, k) such that the 1 2 Z n -grading and Z 2n -grading correspond respectively to the usual refined and unrefined gradings on strands algebras.…”

Section: Basismentioning

confidence: 99%

Singular crossings and Ozsváth–Szabó’s Kauffman-states functor

Manion

2021

Fund. Math.

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Recently, Ozsváth and Szabó introduced some algebraic constructions computing knot Floer homology in the spirit of bordered Floer homology, including a family of algebras B(n) and, for a generator of the braid group on n strands, a certain type of bimodule over B(n). We define analogous bimodules for singular crossings. Our bimodules are motivated by counting holomorphic disks in a bordered sutured version of a Heegaard diagram considered previously by Ozsváth, Stipsicz, and Szabó.

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“…More recently, Ozsváth-Szabó [OSz18, OSz17, OSza, OSzc] have used the ideas of bordered Floer homology to define a new algorithmic method for computing HF K by decomposing a knot into tangles. Their theory has striking computational properties [OSzb], categorifies aspects of the representation theory of U q (gl(1|1)) [Man19], and has surprising connections with other such categorifications [Man17]. We will refer to their theory as the Kauffmanstates functor, since Kauffman states for a knot or tangle projection (equivalently, spanning trees of the Tait graph) play a prominent role.…”

Section: Introductionmentioning

confidence: 99%

Strands algebras and Ozsváth-Szabó's Kauffman-states functor

Manion,

Marengon,

Willis

2019

Preprint

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We define new differential graded algebras A(n, k, S) in the framework of Lipshitz-Ozsváth-Thurston's and Zarev's strands algebras from bordered Floer homology. The algebras A(n, k, S) are meant to be strands models for Ozsváth-Szabó's algebras B(n, k, S); indeed, we exhibit a quasi-isomorphism from B(n, k, S) to A(n, k, S). We also show how Ozsváth-Szabó's gradings on B(n, k, S) arise naturally from the general framework of group-valued gradings on strands algebras. * w unExtend w un additively to any path γ ∈ Path(Γ(n, k, S)).• Let {e 1 , . . . , e n } denote the standard basis of Z n . Define the refined Alexander multigrading on Quiv(Γ(n, k, S), R S ), a grading by 1 2 Z n , by applying the homomorphismn sending τ i and β i to e i 2 to the unrefined Alexander multi-degrees. For a ∈ Quiv(Γ(n, k, S), R S ) homogeneous, let w(a) denote the refined Alexander multidegree of a. Explicitly, for an edge γ of Γ(n, k, S), we haveLet w i (a) denote the coefficient of w(a) on the basis element e i .• Define the single Alexander grading on Quiv(Γ(n, k, S), R S ), a grading by 1 2 Z, by applying the homomorphismto the refined Alexander multi-degrees. Let Alex(a) denote the single Alexander degree of a. We haveExplicitly, for a single edge γ, we have * Alex(γ) = 1/2 if γ has label R i or L i and i / ∈ S * Alex(γ) = −1/2 if γ has label R i or L i and i ∈ S * Alex(γ) = 1 if γ has label U i and i / ∈ S * Alex(γ) = −1 if γ has label U i or C i and i ∈ S.• Define the Maslov grading on Quiv(Γ(n, k, S), R S ), a grading by Z, by declaringfor a path γ in Γ(n, k, S), where # C (γ) is the number of edges in γ labeled C i for some i. Explicitly, for a single edge γ, we haveRemark 2.6. Our use of the words "refined" and "unrefined" follows the standard usage in bordered Floer homology, in contrast with [Man17] (see Section 6 below).Definition 2.7. The dg algebra B(n, k, S) is defined to be Quiv(Γ(n, k, S), R S ), with any of the above three Alexander gradings as an intrinsic grading (preserved by ∂) and the Maslov grading as a homological grading (decreased by 1 by ∂).The above definition is justified by the following theorem.Theorem 2.8 ([MMW19, Corollary 3.14]). The dg algebra B(n, k, S) defined in [OSz18] is isomorphic to Quiv(Γ(n, k, S), R S ).

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

Cited by 12 publications

References 36 publications

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

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

Singular crossings and Ozsváth–Szabó’s Kauffman-states functor

Strands algebras and Ozsváth-Szabó's Kauffman-states functor

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