Heegaard Floer homologies for (+1) surgeries on torus knots

TL;DR

This paper computes the Heegaard Floer homology for (+1) surgeries on torus knots, providing explicit formulas and relating these to known invariants, revealing a notable similarity with (-1) surgeries.

Contribution

It introduces a compact formula for the Heegaard Floer homology of (+1) surgeries on torus knots using semigroup and Dedekind sums, and explores the relation to (-1) surgeries.

Findings

Explicit formula involving Dedekind sums for d-invariant

Relation between (+1) and (-1) surgeries at the level of τ functions

Connection to classical knot invariants like genus and signatures

Abstract

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 on torus knots. This relation is best seen at the level of $\tau$ functions.