Related papers: The sutured Floer homology polytope

Ozsv\'ath and Szab\'o conjectured that knot Floer homology detects fibred knots in $S^3$. We will prove this conjecture for null-homologous knots in arbitrary closed 3--manifolds. Namely, if $K$ is a knot in a closed 3--manifold $Y$, $Y-K$…

For a given smooth $2$-knot in $S^4$, we relate the existence of a smooth Seifert hypersurface of a certain class to the existence of irreducible $ SU(2)$-representations of its knot group. For example, we see that any smooth $2$-knot…

We prove a basic inequality for the d-invariants of a splice of knots in homology spheres. As a result, we are able to prove a new relation on the rank of reduced Floer homology under maps between Seifert fibered homology spheres, improving…

In an earlier paper, we used the absolute grading on Heegaard Floer homology to give restrictions on knots in $S^3$ which admit lens space surgeries. The aim of the present article is to exhibit stronger restrictions on such knots, arising…

In a recent paper, the first author and his collaborator developed a method to compute an upper bound of the dimension of instanton Floer homology via Heegaard Diagrams of 3-manifolds. For a knot inside S3, we further develop an algorithm…

In this article, the authors review what the Floer homology is and what it does in symplectic geometry both in the closed string and in the open string context. In the first case, the authors will explain how the chain level Floer theory…

We calculate the Heegaard Floer homologies$HF^+(M,s) for mapping tori M associated to certain surface diffeomorphisms, where s is any Spin^c structure on M whose first Chern class is non-torsion. Let gamma and delta be a pair of…

In this paper we study the knot Floer homology invariants of the twisted and untwisted Whitehead doubles of an arbitrary knot K. We present a formula for the filtered chain homotopy type of HFK(D(+,K,t)) in terms of the invariants for K,…

This work has two goals. The first is to provide a conceptual introduction to Heegaard Floer homology, the second is to survey the current state of the field, without aiming for completeness. After reviewing the structure of Heegaard Floer…

This paper gives a geometric interpretation of bordered Heegaard Floer homology for manifolds with torus boundary. If $M$ is such a manifold, we show that the type D structure $\widehat{\mathit{CFD}}$ may be viewed as a set of immersed…

We consider a homology sphere $M_n(K_1,K_2)$ presented by two knots $K_1,K_2$ with linking number 1 and framing $(0,n)$. We call the manifold {\it Matsumoto's manifold}. We show that there exists no contractible bound of $M_n(T_{2,3},K_2)$…

We construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes…

This paper establishes a new technique that enables us to access some fundamental structural properties of instanton Floer homology. As an application, we establish, for the first time, a relation between the instanton Floer homology of a…

In this paper, we construct a Hamiltonian Floer theory based invariant called relative symplectic cohomology, which assigns a module over the Novikov ring to compact subsets of closed symplectic manifolds. We show the existence of…

To each knot $K\subset S^3$ one can associated its knot Floer homology $\hat{HFK}(K)$, a finitely generated bigraded abelian group. In general, the nonzero ranks of these homology groups lie on a finite number of slope one lines with…

Given a grid presentation of a knot (or link) K in the three-sphere, we describe a Heegaard diagram for the knot complement in which the Heegaard surface is a torus and all elementary domains are squares. Using this diagram, we obtain a…

In principle, Floer theory can be extended to define homotopy invariants of families of equivalent objects (e.g. Hamiltonian isotopic symplectomorphisms, 3-manifolds, Legendrian knots, etc.) parametrized by a smooth manifold B. The…

We define Floer homology theories for oriented, singular knots in S^3 and show that one of these theories can be defined combinatorially for planar singular knots.

Given a compact, oriented, connected surface $F$, we show that the set of connected sutured manifolds $(M,\gamma)$ with $R_{\pm}(\gamma)\cong F$ is generated by the product sutured manifold $(F,\partial F)\times[0,1]$ through surgery…

We define a version of spectral invariant in the vortex Floer theory for a $G$-Hamiltonian manifold $M$. This defines potentially new (partial) symplectic quasi-morphism and quasi-states when $M//G$ is not semi-positive. We also establish a…