Towards a version of Markov's theorem for ribbon torus-links in ${\mathbb{R}}^4$

Abstract

In classical knot theory, Markov's theorem gives a way of describing all braids with isotopic closures as links in ${\mathbb{R}}^3$. We present a version of Markov's theorem for extended loop braids with closure in $B^3\times S^1$, as a first step towards a Markov's theorem for extended loop braids and ribbon torus-links in ${\mathbb{R}}^4$.