Knots which admit a surgery with simple knot Floer homology groups

We show that if a positive integral surgery on a knot K inside a homology sphere X with Seifert genus g(K) results in an induced knot K_n in X_n(K)=Y which has simple Floer homology, we should have n>=2g(K). Moreover, if X is the standard sphere, the three-manifold Y is a L-space and the Heegaard Floer homology groups of K are determined by its Alexander polynomial

Filtration of Heegaard Floer homology and gluing formulas

We introduce an extra filtration of \CFK(Y,K) and use it in order to obtain formulas for Floer homology of $(Y,K)$, which is obtained from $(Y_i,K_i), i=1,2$ by gluing the knot complements on the framed torus boundaries

Correction to the article: Floer homology and splicing knot complements

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 on the knot Floer homology of $K$ which corresponds to moving the basepoints by one full twist around $K$ is trivial. The correction implicitly involves considering the contribution from this (possibly non-trivial) involution in a number of places

Seifert fibered homology spheres with trivial Heegaard Floer homology

We show that among Seifert fibered integer homology spheres, Poincare sphere (with either orientation) is the only non-trivial example which has trivial Heegaard Floer homology. Together with an earlier result, this shows that if an integer homology sphere has trivial Heegaard Floer homology, then it is a connected sum of a number of Poincare spheres and hyperbolic homology spheres

Embedded curves and Gromov-Witten invariants of three-folds

Associated with a prime homology class $\beta \in P_2(X,\Z)$ (i.e. $\beta=p\alpha$ and $\alpha \in H_2(X,\Z)$ imply $p=1$ or $p$ is an odd prime) on a symplectic three-manifold with vanishing first Chern class, we count the embedded perturbed pseudo-holomorphic curves in $X$ of a fixed genus $g$ to obtain certain integer valued invariants analogous to Gromov-Witten invariants of $X$

Heegaard Floer homologies of pretzel knots

We compute the Ozsv\'ath-Szab\'o Heegaard Floer homology of three stranded pretzel knots

Floer Cohomology of Certain pseudo-Anosov Maps

Floer cohomology is computed for certain elements of the mapping class group of a surface $\Sigma$ of genus $g>1$ which are compositions of positive and negative dehn twists along some loops in $\Sigma$. The computations cover a certain class of pseudo-Anasov maps.Comment: 19 pages, 4 figure

Knots in homology spheres which have simple knot Floer homology are trivial

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.Comment: This paper has been withdrawn by the author. The surgery formulas are used incorrectly for obtaining the main result of the pape

Bordered Floer homology and existence of incompressible tori in homology spheres

Let $K$ denote a knot inside the homology sphere $Y$. The zero-framed longitude of $K$ gives the complement of $K$ in $Y$ the structure of a bordered three-manifold, which may be denoted by $Y(K)$. We compute the quasi-isomorphism type of the bordered Floer complex of $Y(K)$ in terms of the knot Floer complex associated with $K$. As a corollary, we show that if a homology sphere has the same Heegaard Floer homology as $S^3$ it does not contain any incompressible tori. Consequently, if $Y$ is an irreducible homology sphere $L$-space then $Y$ is either $S^3$, or the Poicar\'e sphere $\Sigma(2,3,5)$, or it is hyperbolic

On finiteness and rigidity of J-holomorphic curves in symplectic three-folds

Given a symplectic three-fold $(M,\omega)$ we show that for a generic almost complex structure $J$ which is compatible with $\omega$, there are finitely many $J$-holomorphic curves in $M$ of any genus $g\geq 0$ representing a homology class $\beta$ in \H_2(M,\Z) with $c_1(M).\beta=0$, provided that the divisibility of $\beta$ is at most 4 (i.e. if $\beta=n\alpha$ with $\alpha\in H_2(M,\Z)$ and $n\in \Z$ then $n\leq 4$). Moreover, each such curve is embedded and 4-rigid.Comment: This is a revision of the original submission. The assumption on the homology class is imposed in order to fill the gap in the original versio