Ma-Qiu index, presentation distance, and local moves in knot theory

TL;DR

This paper introduces the Ma-Qiu index as a new invariant that provides lower bounds on presentation distance and Gordian distance in knot theory, unifying various unknotting number bounds.

Contribution

It establishes the Ma-Qiu index as a lower bound for presentation and Gordian distances, linking group invariants to knot unknotting measures.

Findings

Ma-Qiu index bounds presentation distance between groups.

Ma-Qiu index bounds Gordian distance in knot theory.

Unified proof of Nakanishi index bounds for various unknotting numbers.

Abstract

The Ma-Qiu index of a group is the minimum number of normal generators of the commutator subgroup. We show that the Ma-Qiu index gives a lower bound of the presentation distance of two groups, the minimum number of relator replacements to change one group to the other. Since many local moves in knot theory induce relator replacements in knot groups, this shows that the Ma-Qiu index of knot groups gives a lower bound of the Gordian distance based on various local moves. In particular, this gives a unified and simple proof of the Nakanishi index bounds of various unknotting numbers, including virtual or welded knot cases.