Virtual braids from a topological viewpoint

Abstract

Virtual braids are a combinatorial generalization of braids. We present abstract braids as equivalence classes of braid diagrams on a surface, joining two distinguished boundary components. They are identified up to isotopy, compatibility, stability and Reidemeister moves. We show that virtual braids are in a bijective correspondence with abstract braids. Finally we demonstrate that for any abstract braid, its representative of minimal genus is unique up to compatibility and Reidemeister moves. The genus of such a representative is thus an invariant for virtual braids. We also give a complete proof of the fact that there is a bijective correspondence between virtually equivalent virtual braid diagrams and braid-Gauss diagrams.

Unrestricted virtual braids, fused links and other quotients of virtual braid groups

“…In this Section, in order to define unrestricted virtual braid groups, we will first introduce virtual and welded braid groups by simply recalling their group presentation; for other definitions, more intrinsic, see for instance [3,9,17,30] for the virtual case and [8,10,17] for the welded one.…”

“…In F W B n , in addition to relations (9) and (10), we also have relations coming from relations of type (F1), i.e.,…”

A journey through loop braid groups

“…Defining equivalence relations on bGD n , Cisneros proves the existence of a bijection between a quotient of bGD n and V B n [17,Proposition 2.24]. This can be extended to W B n , giving us the last isomorphism of this paper.…”

“…Therefore we need to prove that if two strand diagrams are related by a classical Reidemeister move (R2), (R3), or a welded Reidemeister move (F 1), then their braid Gauss diagram are wReidemeister equivalent via moves (Ω 2 ), (Ω 3 ) and (T C), and viceversa. Moves (Ω 1 ) and (Ω 2 ) are treated in [17,Proposition 2.24]. Let β and β be two welded braids that differ by an (F 1) move, and suppose that the strands involved are a, b and c in {1, .…”

Virtual tangles and fiber functors

“…. One can instead choose an abstract punctured surface on which the non planar graph representing a given virtual knot can be drawn without self-intersection [CdlC15,Kup03,SCKS02]. This point of view leads naturally to a topological interpretation of virtual knotted object, as link or tangle diagrams drawn on surfaces modulo homeomorphisms and certain additional relations (tearing and puncturing) which reflects the fact that the choice of the surface is not unique.…”