Virtual singular braids and links

Abstract

Abstract. Virtual singular braids are generalizations of singular braids and virtual braids. We define the virtual singular braid monoid via generators and relations, and prove Alexander-and Markov-type theorems for virtual singular links. We also show that the virtual singular braid monoid has another presentation with fewer generators.

The L-move and Markov theorems for trivalent braids

“…In Figure 24 we consider the braided portions obtained from a switch move performed on the right hand side of a vertex. It is clear that the braids (1) and (2) in Figure 24 differ by planar isotopy. The case where the two up-arcs are on the left hand side of a Y -type vertex is treated similarly (this can be seen by reflecting the diagrams in Figure 24 across a vertical axis).…”

“…Once again, the braids resulting from either choice differ by a switch move. In Figure 25 we show that this version of the switch move applied on the right hand side of a λ-type vertex does not affect the final braid up to T L-equivalence; specifically, the braids (1) and (2) Figure 24. Checking a switch move on a Y -type vertex with two up-arcs Now consider a Y -type vertex v incident with three up-arcs.…”

“…We remark that the L-move was extended to other diagrammatic situations, including virtual braids [7], virtual singular braids [2], and virtual trivalent braids [3].…”

Diagrammatic representations of knots and links as closed braids

“…Furthermore, the L-move techniques have been used by Vassily Manturov and Hang Wang for formulating a Markovtype theorem for free links [44]. Also, by Carmen Caprau and co-authors for obtaining a braid equivalence for virtual singular braids [11] as well as for virtual trivalent braids [12,13]. Singular knot theory is related to Vassiliev's theory of knot invariants.…”

Algebraic structures among virtual singular braids

“…Due to a theorem by Markov [12], the classification of knots and links is equivalent to certain algebraic properties of classical braids. Similarly, we can study algebraic structures of virtual braids, singular braids, virtual singular braids, and welded braids to classify virtual knots, singular knots, virtual singular knots, and welded knots, respectively (see, for example [1,2,3,4,5,6,7,8,9]).…”