Related papers: Compatibility in Ozsvath-Szabo's bordered HFK via …
We develop the 2-representation theory of the odd one-dimensional super Lie algebra $gl(1|1)^+$ and show it controls the Heegaard-Floer theory of surfaces of Lipshitz, Ozsv\'ath and Thurston. Our main tool is the construction of a tensor…
This is a survey article about knot Floer homology. We present three constructions of this invariant: the original one using holomorphic disks, a combinatorial description using grid diagrams, and a combinatorial description in terms of the…
This is an expansion on my talk at the Geometry and Topology conference at McMaster University, May 2004. We outline a program to relate the Heegaard Floer homologies of Ozsvath-Szabo, and Seiberg-Witten-Floer homologies as defined by…
The purpose of this thesis is to define a "local" version of Ozsv\'{a}th and Szab\'{o}'s Heegaard Floer homology $\operatorname{\widehat{HFL}}$ for links in the 3-dimensional sphere, i.e. a Heegaard Floer homology…
We give a conjectural algebraic description of the Fukaya category of a complexified hyperplane complement, using the algebras defined in arXiv:0905.1335 from the equivariant cohomology of toric varieties. We prove this conjecture for…
For a knot K and its knot Floer complex CFK^-(K), we introduce an algorithm to compute the bordered Floer bimodule of the complement of the knot and its meridian. The grading of the module computes spin^c-summands of a meridional knot in…
We propose a framework for unifying the sl(N) Khovanov-Rozansky homology (for all N) with the knot Floer homology. We argue that this unification should be accomplished by a triply graded homology theory which categorifies the HOMFLY…
We give a combinatorial proof of the quasi-invertibility of $\widehat{CFDD}(\mathbb{I}_\mathcal{Z})$ in bordered Heegaard Floer homology, which implies a Koszul self-duality on the dg-algebra $\mathcal{A}(\mathcal{Z})$, for each pointed…
The Khovanov homology of a link in $S^3$ and the Heegaard Floer homology of its branched double cover are related through a spectral sequence constructed by Ozsv\'ath and Szab\'o. This spectral sequence has topological applications but is…
We give a partial characterization of bordered Floer homology in terms of sutured Floer homology. The bordered algebra and modules are direct sums of certain sutured Floer complexes. The algebra multiplication and algebra action correspond…
Bordered Floer homology is an invariant for 3-manifolds with boundary, defined by the authors in 2008. It extends the Heegaard Floer homology of closed 3-manifolds, defined in earlier work of Zolt\'an Szab\'o and the second author. In…
We give a generators-and-relations description of differential graded algebras recently introduced by Ozsv\'ath and Szab\'o for the computation of knot Floer homology. We also compute the homology of these algebras and determine when they…
The Burau representation of braid groups and knot Floer homology share a link to the Fox calculus. We make this connection explicit, with the following outcome: if $B$ is the full Burau matrix of any braid, and $A$ is any square submatrix…
We show that a decorated knot concordance $\mathcal{C}$ from $K_0$ to $K_1$ induces an $\mathbb{F}[U]$-module homomorphism \[G_{\mathcal{C}}: HFK^{-}(-S^3,K_0) \to HFK^{-}(-S^3,K_1)\] which preserves the Alexander and absolute…
In 2005 Dunfield, Gukov and Rasmussen conjectured an existence of the spectral sequence from the reduced triply graded Khovanov-Rozansky homology of a knot to its knot Floer homology defined by Ozsv\'ath and Szab\'o. The main result of this…
We give new obstructions to the module structures arising in Heegaard Floer homology. As a corollary, we characterize the possible modules arising as the Heegaard Floer homology of an integer homology sphere with one-dimensional reduced…
We use involutive Heegaard Floer homology to extend the Ozsv\'ath-Szab\'o branched double cover spectral sequence relating a version of Khovanov homology and the Heegaard Floer homology of branched double covers. Our main tools are…
Using bordered Floer theory, we give a combinatorial construction and proof of invariance for the hat version of Heegaard Floer homology. As a part of the proof, we also establish combinatorially the invariance of the linear-categorical…
We introduce a simple combinatorial method for computing all versions of the knot Floer homology of the preimage of a two-bridge knot K(p,q) inside its double-branched cover, -L(p,q). The 4-pointed genus 1 Heegaard diagram we obtain looks…
We show that Khovanov homology (and its sl(3) variant) can be understood in the context of higher representation theory. Specifically, we show that the combinatorially defined foam constructions of these theories arise as a family of…