Heegaard Floer homology and Morse surgery

Extensive rewrite. Tables and proofs have been reformatted and/or rewritten for clarity.

We compute the Heegaard Floer homology of $S^3_1(K)$ (the (+1) surgery on the torus knot $T_{p,q}$) in terms of the semigroup generated by $p$ and $q$, and we find a compact formula (involving Dedekind sums) for the corresponding Ozsvath--Szabo d-invariant. We relate the result to known knot invariants of $T_{p,q}$ as the genus and the Levine--Tristram signatures. Furthermore, we emphasize the striking resemblance between Heegaard Floer homologies of (+1) and (-1) surgeries o...

In this paper we study the knot Floer homology of a subfamily of twisted $(p, q)$ torus knots where $q \equiv\pm1$ (mod $p$). Specifically, we classify the knots in this subfamily that admit L-space surgeries. To do calculations, we use the fact that these knots are $(1, 1)$ knots and, therefore, admit a genus one Heegaard diagram.

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. This exact triangle is different from the one which appeared in Ozsv\'{a}th and Szab\'{o}'s original proof. This exact triangle simplifies a number of technical aspects in their proofs and also allows us to prove several new results. A first a...

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.

Using the correction terms in Heegaard Floer homology, we prove that if a knot in $S^3$ admits a positive integral $\mathbf{T}$-, $\mathbf{O}$- or $\mathbf{I}$-type surgery, it must have the same knot Floer homology as one of the knots given in our complete list, and the resulting manifold is orientation-preservingly homeomorphic to the $p$-surgery on the corresponding knot.

We give a precise description of splicing formulas from a previous paper in terms of knot Floer complex associated with a knot in homology sphere.

We define sutured Heegaard diagrams for null-homologous knots in 3-manifolds. These diagrams are useful for computing the knot Floer homology at the top filtration level. As an application, we give a formula for the knot Floer homology of a Murasugi sum. Our result echoes Gabai's earlier works. We also show that for so-called 'semifibred' satellite knots, the top filtration term of the knot Floer homology is isomorphic to the counterpart of the companion.

We use an algorithm by Ozsvath and Szabo to find closed formulae for the ranks of the hat version of the Heegaard Floer homology groups for non-zero Dehn surgeries on knots in the 3-sphere. As applications we provide new bounds on the number of distinct ranks of the Heegaard Floer groups a Dehn surgery can have. These in turn give a new lower bound on the rational Dehn surgery genus of a rational homology 3-sphere. We also provide novel obstructions for a knot to be a potenti...

This largely technical paper is divided into two parts: part I: An account of P. Ozsvath and Z. Szabo's construction of the link surgery spectral sequence. There are no new results here, but this part slightly modifies and expands their proof and is included as an aid to part II. part II: Some modifications of the spectral sequence suitable for knot Floer homology; an exposition of the invariance of the spectral sequence, under a suitable equivalence, from alterations of the ...