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

Abstract

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.

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

“…We review some dg algebras introduced by Ozsváth-Szabó in [OSz18]. The algebra B(n) mentioned above is a direct sum of algebras B(n, k) for 0 ≤ k ≤ n. The following generators-and-relations description of B(n, k) is shown in [MMW19a] to be equivalent to the definition given in [OSz18].…”

“…Ozsváth-Szabó give such a basis for B(n, k) in [OSz18, Proposition 3.7]. The basis elements can be described in terms of quiver generators as in [MMW19a,Corollary 4.12]; we do this for n = 2 below.…”

“…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].…”

“…The first symmetry, which they call R, gives dg ring endomorphisms of B(n, k) and B ! (n, k) with R 2 = id (the symmetry R was denoted ρ in [MMW19a,MMW19b]). It acts as a nontrivial involution on the primitive idempotents; explicitly, R restricts to a map from I(n, k) to itself sending R(I x ) = I y where y := {n − i | i ∈ x}.…”

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

“…The bordered strands algebra A(n, k, S) and Ozsváth-Szabó's algebra B(n, k, S) are in fact closely related to each other. Using the generators-and-relations description of B(n, k, S) from [MMW19], we define a dg algebra homomorphism Φ : B(n, k, S) → A(n, k, S) and prove the following result.…”