On knot Floer homology and lens space surgeries

TL;DR

This paper strengthens restrictions on knots in S^3 that admit lens space surgeries using knot Floer homology, leading to classifications and bounds related to such surgeries and their resulting manifolds.

Contribution

It introduces new restrictions based on knot Floer homology, including Alexander polynomial coefficients and classifications of lens spaces from surgeries.

Findings

All non-zero Alexander polynomial coefficients are ±1.

Classified lens spaces obtainable from surgeries with |p| ≤ 1500.

Provided bounds on four-ball genera for knots with lens space surgeries.

Abstract

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 from knot Floer homology. One consequence is that all the non-zero coefficients of the Alexander polynomial of such a knot are $\pm 1$. This information in turn can be used to prove that certain lens spaces are not obtained as integral surgeries on knots. In fact, combining our results with constructions of Berge, we classify lens spaces $L(p,q)$ which arise as integral surgeries on knots in $S^3$ with $|p|\leq 1500$. Other applications include bounds on the four-ball genera of knots admitting lens space surgeries (which are sharp for Berge's knots), and a constraint on three-manifolds obtained as integer surgeries on alternating knots, which is closely to related to a theorem of Delman and Roberts.