Algebraic structures among virtual singular braids

Abstract

We show that the virtual singular braid monoid on n strands embeds in a group V SGn, which we call the virtual singular braid group on n strands. The group V SGn contains a normal subgroup V SP Gn of virtual singular pure braids. We show that V SGn is a semi-direct product of V SP Gn and the symmetric group Sn. We provide a presentation for V SP Gn via generators and relations. We also represent V SP Gn as a semi-direct product of n − 1 subgroups and study the structures of these subgroups. These results yield a normal form of words in the virtual singular braid group.

On virtual singular braid groups

“…(CR) Commuting relations: [10,Theorem 4] we conclude that the groups B n , V B n and SG n are contained in the virtual singular braid group V SG n .…”

“…Now we need to check that for the other four triples the map ϕ ε 1 , ε 2 , ε 3 is a homomorphism. It is clear for (ε 1 , ε 2 , ε 3 ) equal to (0, 0, 0) since we get the trivial map, and also for Page 6] and its kernel is the so-called virtual singular pure braid group, see [10,Definition 5]. Finally, that ϕ ε 1 , ε 2 , ε 3 is a homomorphism for (ε 1 , ε 2 , ε 3 ) equal to (1, 0, 1) or (0, 0, 1) follows from a simple computation using the canonical presentations of the groups involved.…”

“…In [10, Definition 3] was defined an abstract group called the virtual singular braid group such that the virtual singular braid monoid V SB n (defined in [9]) embeds in it. We shall use the presentation as stated in [10,. We note that the relations of the groups B n , V B n and SG n given in their respective presentations (see Definition 1 and 2) appears in in the following presentation.…”

“…A similar topological interpretation are given for virtual braid groups, singular braid groups and virtual singular braid groups. We illustrate here the case of the virtual singular braid group (see also [10] and [12]). These elements have three types of crossings, classical (σ i ), singular (τ i ) and virtual (v i ), which correspond to the generators illustrated in Figures 1 and 2.…”

“…In [9] the authors proved an Alexander and Markov Theorem for virtual singular braids and gave two presentations for the monoid of virtual singular braids, denoted by V SB n . In a more recent paper [10] Caprau and Yeung showed that the monoid V SB n embeds in the group V SG n called the virtual singular braid group on n strands. They also gave a presentation of the virtual singular pure braid group V SP G n and showed that V SG n is a semi-direct product of V SP G n and the symmetric group S n .…”