Untwisting information from Heegaard Floer homology

TL;DR

This paper explores the untwisting number, a generalization of the unknotting number, using Heegaard Floer homology to obstruct certain knots from being simplified with minimal twists.

Contribution

It introduces new Heegaard Floer correction term conditions for the untwisting number and demonstrates their effectiveness in obstructing low untwisting numbers for specific knots.

Findings

Heegaard Floer correction terms can obstruct untwisting number one.

The Montesinos trick does not extend to the untwisting number.

The Ozsváth-Szabó tau and Rasmussen s invariants differentiate p- and q-untwisting numbers.

Abstract

The unknotting number of a knot is the minimum number of crossings one must change to turn that knot into the unknot. We work with a generalization of unknotting number due to Mathieu-Domergue, which we call the untwisting number. The p-untwisting number is the minimum number (over all diagrams of a knot) of full twists on at most 2p strands of a knot, with half of the strands oriented in each direction, necessary to transform that knot into the unknot. In previous work, we showed that the unknotting and untwisting numbers can be arbitrarily different. In this paper, we show that a common route for obstructing low unknotting number, the Montesinos trick, does not generalize to the untwisting number. However, we use a different approach to get conditions on the Heegaard Floer correction terms of the branched double cover of a knot with untwisting number one. This allows us to obstruct several 10 and 11-crossing knots from being unknotted by a single positive or negative twist. We also use the Ozsv\'ath-Szab\'o tau invariant and the Rasmussen s invariant to differentiate between the p- and q-untwisting numbers for certain p,q > 1.