Unknotting via null-homologous twists and multi-twists

TL;DR

This paper explores the untwisting number of knots, a measure of how many null-homologous twists are needed to unknot a knot, and compares it with related invariants like the surgery description number, revealing their differences and bounds.

Contribution

It introduces the concept of the untwisting number as a generalization of crossing changes and compares it with the surgery description number, establishing bounds and differences for infinitely many knots.

Findings

Untwisting number is at most twice the surgery description number plus one.

Untwisting and surgery description numbers differ for infinitely many knots.

The untwisting number bounds the topological 4-genus, generalizing the unknotting number.

Abstract

The untwisting number of a knot K is the minimum number of null-homologous twists required to convert K to the unknot. Such a twist can be viewed as a generalization of a crossing change, since a classical crossing change can be effected by a null-homologous twist on 2 strands. While the unknotting number gives an upper bound on the smooth 4-genus, the untwisting number gives an upper bound on the topological 4-genus. The surgery description number, which allows multiple null-homologous twists in a single twisting region to count as one operation, lies between the topological 4-genus and the untwisting number. We show that the untwisting and surgery description numbers are different for infinitely many knots, though we also find that the untwisting number is at most twice the surgery description number plus 1.