The Alexander and Markov theorems for strongly involutive links

TL;DR

This paper extends the classical Alexander and Markov theorems to strongly involutive links in S3, introducing an equivariant closure map for palindromic braids and establishing their relation through equivariant Markov moves.

Contribution

It develops an equivariant closure map for palindromic braids and proves its surjectivity and the equivalence of links via equivariant Markov moves for strongly involutive links.

Findings

The equivariant closure map is surjective up to link equivalence.

Pairs of palindromic braids with the same closure are related by equivariant Markov moves.

The results specialize classical theorems to strongly involutive links in S3.

Abstract

The Alexander theorem (1923) and the Markov theorem (1936) are two classical results in knot theory that show respectively that every link is the closure of a braid and that braids that have the same closure are related by a finite number of operations called Markov moves. This paper presents specialized versions of these two classical theorems for a class of links in S3 preserved by an involution, that we call strongly involutive links. When connected, these links are known as strongly invertible knots, and have been extensively studied. We develop an equivariant closure map that, given two palindromic braids, produces a strongly involutive link. We demonstrate that this map is surjective up to equivalence of strongly involutive links. Furthermore, we establish that pairs of palindromic braids that have the same equivariant closure are related by an equivariant version of the original Markov moves.