Reidemeister and movie moves for involutive links

TL;DR

This paper develops an equivariant framework for studying involutive links in $S^3$, introducing 39 movie moves and analyzing singularities to understand equivariant cobordisms and Reidemeister moves.

Contribution

It establishes an equivariant analogue of Carter and Saito's work, providing a complete set of movie moves for involutive links and a singularity-theoretic proof of the equivariant Reidemeister theorem.

Findings

39 equivariant movie moves suffice for involutive links

Singularity-theoretic proof of the equivariant Reidemeister theorem

Analysis of codimension 2 singularities in equivariant maps

Abstract

An involutive link is a link which is invariant under the standard rotation by 180 degrees in $S^3$. We establish an equivariant analogue of the work of Carter and Saito aimed at studying equivariant cobordisms between involutive links. This gives a set of $39$ equivariant movie moves that suffice to go between any two movie presentations of a pair of equivariantly isotopic cobordisms. Along the way, we give a singularity-theoretic proof of the equivariant Reidemeister theorem and study loops of equivariant Reidemeister moves. Our approach proceeds by analyzing codimension $2$ singularities of equivariant maps from $S^1$ to $\mathbb{R}^2$, as well as utilizing embedded equivariant Morse theory.