Unknotting information from Heegaard Floer homology

TL;DR

This paper employs Heegaard Floer homology to establish bounds on unknotting numbers, extending previous obstructions, and determines the unknotting numbers for several prime knots with crossing number nine or less.

Contribution

It generalizes Ozsvath and Szabo's unknotting number obstruction using a refined Montesinos' theorem and completes the unknotting number table for prime knots up to nine crossings.

Findings

Determined unknotting numbers for specific prime knots.

Extended the obstruction method to a broader class of knots.

Provided a complete table for prime knots with ≤9 crossings.

Abstract

We use Heegaard Floer homology to obtain bounds on unknotting numbers. This is a generalisation of Ozsvath and Szabo's obstruction to unknotting number one. We determine the unknotting numbers of 9_10, 9_13, 9_35, 9_38, 10_53, 10_101 and 10_120; this completes the table of unknotting numbers for prime knots with crossing number nine or less. Our obstruction uses a refined version of Montesinos' theorem which gives a Dehn surgery description of the branched double cover of a knot.