A Markov's theorem for extended welded braids and links

Abstract

Extended welded links are a generalization of Fenn, Rimányi, and Rourke's welded links. Their braided counterpart are extended welded braids, which are closely related to ribbon braids and loop braids. In this paper we prove versions of Alexander and Markov's theorems for extended welded braids and links, following Kamada's approach to the case of welded objects.

Multi-Skein Invariants for Welded and Extended Welded Knots and Links

“…The complete family of moves allowed in the theory of extended welded links (and corresponding implicitly to the relations in the extended welded braid groups) are shown in Figure 1, where a wen is indicated by an angled mark in the location of the crossing. Damiani [3] proved an analogue of Markov's theorem relating the extended welded braid groups to extended welded links in the anticipated way. The interpretation of welded and extended welded braids as motions of loops in R 3 suggests that surfaces ambient isotopic to the closures of geometric loop braids, with or without wens, will form an interesting class of geometric objects in R 4 .…”

“…Once this move is included, what had been called virtual crossings should be more properly called welded crossings. Following Damiani [3], we denote this move by F1. Several results on classical links and braids generalize nicely to welded links and braids; in particular, Kauffman and Lambropoulou [9] proved analogues of the Alexander and Markov theorems for welded links.…”

“…The introduction of wen marks (cf. Damiani [3,4]) obviates this problem, as a horizontal mirror image will always be equivalent under extended welded Reidemeister moves to the original link. Moreover, there are other non-welded-equivalent pairs of welded links which become equivalent once it is possible to introduce a pair of wen marks on a component and then move one along the component until they can again be cancelled.…”

“…By virtue of Damiani's [3] Markov theorem, it will suffice to verify invariance under the moves in the braid-like position shown in Figure 1, there being for instance no need to consider directly the second Reidemeister move with one strand oriented up, the other oriented down.…”