Related papers: A Simplification of Combinatorial Link Floer Homol…

We introduce a refinement of the Ozsvath-Szabo complex associated to a balanced sutured manifold $(X,\tau)$ by Juhasz. An algebra $A_\tau$ is associated to the boundary of a sutured manifold and a filtration of its generators by…

Using lattice homology, we give an explicit combinatorial description of the Seiberg-Witten-Floer spectrum $\mathit{SWF}(Y)$ for $Y$ an almost-rational plumbed homology sphere. This class of manifolds includes all Seifert fibered rational…

Quasi-alternating links are homologically thin for both Khovanov homology and knot Floer homology. We show that every quasi-alternating link gives rise to an infinite family of quasi-alternating links obtained by replacing a crossing with…

We develop a skein exact sequence for knot Floer homology, involving singular knots. This leads to an explicit, algebraic description of knot Floer homology in terms of a braid projection of the knot.

We define an annular concordance invariant and study its properties. When specialized to braids, this invariant gives bounds on band rank. We introduce a modified chain complex to reformulate the invariant. Then, by focusing on a special…

This note corrects the mistakes in the splicing formulas of the paper "Floer homology and splicing knot complements". The mistakes are the result of the incorrect assumption that for a knot $K$ inside a homology sphere $Y$, the involution…

If L is an oriented link with $n$ components, then the rank of its Khovanov homology is at least $2^n$. We classify all the links whose Khovanov homology with Z/2-coefficients achieves this lower bound, and show that such links can be…

We study the Ozsv\'{a}th-Szab\'{o}-Thurston transverse invariant in combinatorial link Floer homology for certain transverse cables $\mathscr{L}_{p,q}$ of transverse link $L$ in $S^3$. Transverse cables $\mathscr{L}_{p,q}$ are constructed…

In this paper, we study various homology cobordism invariants for Seifert fibered integral homology 3-spheres derived from Heegaard Floer homology. Our main tool is lattice homology, a combinatorial invariant defined by Ozsv\'ath-Szab\'o…

Khovanov-Floer theories are a class of homological link invariants which admit spectral sequences from Khovanov homology. They include Khovanov homology, Szab{\'o}'s geometric link homology, singular instanton homology, and various Floer…

In this paper, we compute the symplectic Floer homology of the figure eight knot. This provides first nontrivial knot with trivial symplectic Floer homology.

Similar to knots in S^3, any knot in a lens space has a grid diagram from which one can combinatorially compute all of its knot Floer homology invariants. We give an explicit description of the generators, differentials, and rational Maslov…

In this paper we calculate Ozsv\'ath-Szab\'o Floer homology group HF+ for Brieskorn spheres \Sigma(2,2n + 1,4n + 3).

Inspired by the $S^n$ colored version of Khovanov and Khovanov-Rozansky homology, we define a colored version of knot Floer homology by studying the colimit of a directed system of link Floer homology with infinite full twists.…

We show that if K is a non-trivial knot inside a homology sphere X, the rank of the knot Floer homology group associated with K is strictly bigger than the rank of the Heegaard Floer homology group associated with X.

We show that link Floer homology detects the Thurston norm of a link complement. As an application, we show that the Thurston polytope of an alternating link is dual to the Newton polytope of its multi-variable Alexander polynomial. To…

We define a link lattice complex for plumbed links, generalizing constructions of Ozsv\'ath, Stipsicz and Szab\'o, and of Gorsky and N\'emethi. We prove that for all plumbed links in rational homology 3-spheres, the link lattice complex is…

We will prove that, for a $2$ or $3$ component $L$-space link, $HFL^-$ is completely determined by the multi-variable Alexander polynomial of all the sub-links of $L$, as well as the pairwise linking numbers of all the components of $L$. We…

In the present work we generalize the construction of monopole Floer homology due to Kronheimer and Mrowka to the case of a gradient flow with Morse-Bott singularities. Focusing then on the special case of a three-manifold equipped with a…

In the first part of the Thesis, we reformulate the Murakami-Ohtsuki-Yamada state-sum description of the level n Jones polynomial of an oriented link in terms of a suitable braided monoidal category whose morphisms are Q[q, q-1] s-linear…