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 R 3 . We present a version of Markov's theorem for extended loop braids with closure in B 3 ×S 1 , as a first step towards a Markov's theorem for extended loop braids and ribbon torus-links in R 4 .

Milnor invariants of braids and welded braids up to homotopy

“…In the meantime, the interest for welded knotted objects had grown, as the link between welded diagrams, four-dimensional topology and automorphisms of the free group had become more apparent [FRR97,Sat00,BWC07]; see [Dam17] for a survey of welded braids. In recent years, the study of these objects has been flourishing; see for instance [Kam07,BB14,Aud16,NNSY18,MY19,Dam19]. In particular, link-homotopy for these objects (corresponding to self-virtualization moves in welded diagrams) has been the focus of several recent papers [ABMW17, AMW17,AM19].…”