The Alexander and Markov theorems for strongly involutive links

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 specialised versions of these two classical theorems for a class of links in $S^3$ 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.