Mapping cone formula in Heegaard Floer homology and Dehn surgery on knots in $S^3$

TL;DR

This paper provides an explicit formula for the Heegaard Floer homology of Dehn surgeries on knots in $S^3$, enabling new results on knot surgeries and restrictions on Seifert fibred surgeries.

Contribution

It introduces a concrete formula linking Heegaard Floer homology of surgered manifolds to knot invariants, improving understanding of knot surgeries.

Findings

Finiteness of alternating knots producing a fixed manifold via surgery

Lower bounds on knot genus based on resulting manifold

New restrictions on Seifert fibred surgeries

Abstract

We write down an explicit formula for the $+$ version of the Heegaard Floer homology (as an absolutely graded vector space over an arbitrary field) of the results of Dehn surgery on a knot $K$ in $S^3$ in terms of homological data derived from $CFK^{\infty}(K)$. This allows us to prove some results about Dehn surgery on knots in $S^3$. In particular, we show that for a fixed manifold there are only finitely many alternating knots that can produce it by surgery. This is an improvement on a recent result by Lackenby and Purcell. We also derive a lower bound on the genus of knots depending on the manifold they give by surgery. Some new restrictions on Seifert fibred surgery are also presented.