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A bound on the equivariant unknotting number

Sarah Zampa

Abstract

We study how the equivariant signature of strongly invertible knots changes when one of the Boyle-Chen equivariant unknotting moves is applied. It follows form our results that the absolute value of the equivariant signature introduced by Alfieri-Boyle gives a lower bound to three times the equivariant unknotting number.

A bound on the equivariant unknotting number

Abstract

We study how the equivariant signature of strongly invertible knots changes when one of the Boyle-Chen equivariant unknotting moves is applied. It follows form our results that the absolute value of the equivariant signature introduced by Alfieri-Boyle gives a lower bound to three times the equivariant unknotting number.

Paper Structure

This paper contains 9 sections, 12 theorems, 54 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Background material
  3. Admissible diagrams
  4. The equivariant signature and the Goeritz matrix
  5. Proof of Theorem \ref{['mainthmB']}
  6. Proof of Theorem \ref{['mainthmC']}
  7. Proof of Theorem \ref{['mainthmA']} in the case of a type A1 move
  8. Proof of Theorem \ref{['mainthmA']} in the case of a type A2 move
  9. Examples

Key Result

Theorem 1

Let $K$ be a directed strongly invertible knot, and consider a homotopy from $K$ to the unknot which never crosses the direction. Then, $\lvert\widetilde{\sigma}(K)\rvert/3$ gives a lower bound to the number of transverse self-intersections.

Figures (10)

  • Figure 1: Examples of type A, B and C moves applied to $6_1$; we can see that $\widetilde{u}_A(6_1)=\widetilde{u}_B(6_1)=1$.
  • Figure 2: A move showing that one can always make a diagram admissible.
  • Figure 3: The incidence number $\eta(c)=\pm1$ associated to a crossing $c$.
  • Figure 4: Two different type C moves: a positive move (left) and a negative move (right).
  • Figure 5: The resolution of the two new crossings, giving a split link.
  • ...and 5 more figures

Theorems & Definitions (31)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Corollary 5
  • Theorem 6
  • Corollary 7
  • Definition 8
  • Proposition 9
  • proof
  • ...and 21 more