The untwisting number of a knot

TL;DR

This paper introduces the untwisting number, a generalization of the unknotting number involving strand twists, and explores its relationship with the algebraic unknotting number, revealing cases of significant difference.

Contribution

It defines the untwisting number, proves its algebraic version equals the algebraic unknotting number, and demonstrates large disparities between these measures for certain knots.

Findings

Algebraic untwisting number equals algebraic unknotting number.

Existence of knots with arbitrarily large difference between unknotting and untwisting numbers.

The difference persists even with restrictions on the number of strands twisted.

Abstract

The unknotting number of a knot is the minimum number of crossings one must change to turn that knot into the unknot. The algebraic unknotting number is the minimum number of crossing changes needed to transform a knot into an Alexander polynomial-one knot. We work with a generalization of unknotting number due to Mathieu-Domergue, which we call the untwisting number. The untwisting number is the minimum number (over all diagrams of a knot) of right- or left-handed twists on even numbers of strands of a knot, with half of the strands oriented in each direction, necessary to transform that knot into the unknot. We show that the algebraic untwisting number is equal to the algebraic unknotting number. However, we also exhibit several families of knots for which the difference between the unknotting and untwisting numbers is arbitrarily large, even when we only allow twists on a fixed number of strands or fewer.