Related papers: Invariance and the knot Floer cube of resolutions

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 continue our study of the integer-valued knot invariants $\nu^\sharp(K)$ and $r_0(K)$, which together determine the dimensions of the framed instanton homologies of all nonzero Dehn surgeries on $K$. We first establish a "conjugation"…

We study equivariant Seiberg-Witten Floer theory of rational homology $3$-spheres in the special case where the group action is given by an involution. The case of involutions deserves special attention because we can couple the involution…

We explain how to adapt the methods of Abouzaid-McLean-Smith to the setting of Hamiltonian Floer theory. We develop a language around equivariant ``$\langle k \rangle$-manifolds'', which are a type of manifold-with-corners that suffices to…

Given a contact structure on a closed, oriented three-manifold $Y$, we describe an invariant which takes values in the three-manifold's Floer homology $\HFa$. This invariant vanishes for overtwisted contact structures and is non-zero for…

We obtain a formula for the Heegaard Floer homology (hat theory) of the three-manifold $Y(K_1,K_2)$ obtained by splicing the complements of the knots $K_i\subset Y_i$, $i=1,2$, in terms of the knot Floer homology of $K_1$ and $K_2$. We also…

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…

If $K$ is a fibered knot in a closed, oriented $3$--manifold $Y$ with fiber $F$, and $\widehat{HFK}(Y,K,[F], g(F)-1;\mathbb Z/2\mathbb Z)$ has rank $r$, then the monodromy of $K$ is freely isotopic to a diffeomorphism with at most $r-1$…

We extend knot contact homology to a theory over the ring $\mathbb{Z}[\lambda^{\pm 1},\mu^{\pm 1}]$, with the invariant given topologically and combinatorially. The improved invariant, which is defined for framed knots in $S^3$ and can be…

We establish a surgery exact triangle for involutive Heegaard Floer homology by using a doubling model of the involution. We use this exact triangle to give an involutive version of Ozsv\'ath-Szab\'o's mapping cone formula for knot surgery.…

We give several new perspectives on the Heegaard Floer Dehn surgery formulas of Manolescu, Ozsv\'{a}th and Szab\'{o}. Our main result is a new exact triangle in the Fukaya category of the torus which gives a new proof of these formulas.…

We define homology of ternary algebras satisfying axioms derived from particle scattering or, equivalently, from the third Reidemeister move. We show that ternary quasigroups satisfying these axioms appear naturally in invariants of…

O. Plamenevskaya associated to each transverse knot K an element of the Khovanov homology of K. In this paper, we give two refinements of Plamenevskaya's invariant, one valued in Bar-Natan's deformation of the Khovanov complex and another…

This note contains two remarks about the application of the d-invariant in Heegaard Floer homology and Donaldson's diagonalization theorem to knot theory. The first is the equivalence of two obstructions they give to a 2-bridge knot being…

Let M be a closed oriented 3-manifold with first Betti number one. Its equivariant linking pairing may be seen as a two-dimensional cohomology class in an appropriate infinite cyclic covering of the space of ordered pairs of distinct points…

We define a concordance invariant, epsilon(K), associated to the knot Floer complex of K, and give a formula for the Ozsv\'ath-Szab\'o concordance invariant tau of K_{p,q}, the (p,q)-cable of a knot K, in terms of p, q, tau(K), and…

In this article, we focus on the invariance property of Morse homology on noncompact manifolds. We expect to apply outcomes of this article to several types of Floer homology, thus we define Morse homology purely axiomatically and…

We introduce the notion of a Khovanov-Floer theory. Roughly, such a theory assigns a filtered chain complex over Z/2 to a link diagram such that (1) the E_2 page of the resulting spectral sequence is naturally isomorphic to the Khovanov…

Besides offering a friendly introduction to knot homologies and quantum curves, the goal of these lectures is to review some of the concrete predictions that follow from the physical interpretation of knot homologies. In particular, this…

From a handlebody-theoretic perspective, the simplest compact, contractible 4-manifolds, other than the 4-ball, are Mazur manifolds. We produce the first pairs of Mazur manifolds that are homeomorphic but not diffeomorphic. Our…