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.

We extend the definition of Khovanov-Lee homology to links in connected sums of S 1 × S 2 's, and construct a Rasmussen-type invariant for null-homologous links in these manifolds. For certain links in S 1 ×S 2 , we compute the invariant by reinterpreting it in terms of Hochschild homology. As applications, we prove inequalities relating the Rasmussen-type invariant to the genus of surfaces with boundary in the following four-manifolds: B 2 × S 2 , S 1 × B 3 , CP 2 , and various connected sums and boundary sums of these. We deduce that Rasmussen's invariant also gives genus bounds for surfaces inside homotopy 4-balls obtained from B 4 by Gluck twists. Therefore, it cannot be used to prove that such homotopy 4-balls are non-standard. Contents1. Introduction 2. Lee's deformation in # r (S 1 × S 2 ) 2.1. Notation and conventions 2.2. The deformed complex and Lee homology 2.3. Behavior under diffeomorphisms 2.4. Lee generators in KC Lee (T ∞ n ) and KC Lee (D) 3. A generalization of Rasmussen's invariant 3.1.

We modify the construction of knot Floer homology to produce a one-parameter family of homologies tHFK for knots in S 3. These invariants can be used to give homomorphisms from the smooth concordance group C to Z, giving bounds on the four-ball genus and the concordance genus of knots. We give some applications of these homomorphisms.

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The Knight Move Conjecture claims that the Khovanov homology of any knot decomposes as direct sums of some "knight move" pairs and a single "pawn move" pair. This is true for instance whenever the Lee spectral sequence from Khovanov homology to Q 2 converges on the second page, as it does for all alternating knots and knots with unknotting number at most 2. We present a counterexample to the Knight Move Conjecture. For this knot, the Lee spectral sequence admits a nontrivial differential of bidegree (1, 8).arXiv:1809.09769v2 [math.GT]