Heegaard Floer groups of Dehn surgeries

TL;DR

This paper derives explicit formulas for Heegaard Floer homology ranks of non-zero Dehn surgeries on knots, providing bounds and obstructions relevant to knot theory and 3-manifold topology.

Contribution

It introduces a new algorithmic approach to compute Heegaard Floer groups for Dehn surgeries, offering novel bounds and obstructions in knot theory.

Findings

Derived closed formulas for Heegaard Floer homology ranks

Established new bounds on the ranks of Heegaard Floer groups

Provided obstructions related to the Cabling Conjecture

Abstract

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 potential counterexample to the Cabling Conjecture.