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

Tetsuya Ito

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.

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

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.
Paper Structure (1 section, 8 theorems, 10 equations)

This paper contains 1 section, 8 theorems, 10 equations.

Table of Contents

  1. Acknowledgements

Key Result

Theorem 1

Let $G=\langle S \: | \: R \rangle$ and $G'=\langle S \: | \: R' \rangle$ be two (possibly infinite) presentations of groups with the same set of generators. Let $R_0 =R \cap R'$, $R_G = R \setminus R_0$, $R_{G'}=R' \setminus R_0$, and let $\pi_G:\langle S \rangle \rightarrow G$ be the projection ma

Theorems & Definitions (17)

  • Definition 1: Ma-Qiu index
  • Theorem 1
  • proof
  • Corollary 2
  • proof
  • Corollary 3
  • Definition 2: (Null-homologous) relator replacement
  • Definition 3: Presentation distance
  • Corollary 4
  • Definition 4: Proper $n$-tangle replacement
  • ...and 7 more