Let Bn(RP ) (respectively Pn(RP )) denote the braid group (respectively pure braid group) on n strings of the real projective plane RP 2 . In this paper we study these braid groups, in particular the associated pure braid group short exact sequence of Fadell and Neuwirth, their torsion elements and the roots of the ‘full twist’ braid. Our main results may be summarised as follows: first, the pu...

Motivated by the work of Birman about relationship between mapping class groups and braid groups, authors discuss orbit group equivariant on closed surface M with a free proper action in this paper. Their construction is based exact sequence given fibration $${\cal F}_0^GM \to F\left( {M/G,n} \right)$$ . The conclusion closely connected quotient space. Comparing situation without action, there ...

In this paper we prove a Markov Theorem for the virtual braid group and for some analogs of this structure. The virtual braid group is the natural companion to the category of virtual knots, just as the Artin braid group is to classical knots and links. In classical knot theory the braid group gives a fundamental algebraic structure associated with knots. The Alexander Theorem tells us that eve...

We argue that various braid group actions on triangulated categories should be extended to projective actions of the category of braid cobordisms and illustrate how this works in examples. We also construct actions of both the affine braid group and the braid cobordism category on the derived category of coherent sheaves on the cotangent bundle to the full flag variety.

We show several geometric and algebraic aspects of a necklace: a link composed with a core circle and a series of circles linked to this core. We first prove that the fundamental group of the configuration space of necklaces (that we will call braid group of a necklace) is isomorphic to the braid group over an annulus quotiented by the square of the center. We then define braid groups of neckla...

Braid groups were first introduced by Emil Artin in 1925. First cryptosystem, using Braid groups as a platform was discovered by Anshel et al in 2001. After the publication of this paper several cryptosystems on Braid groups had been designed. In this paper we have proposed a tripartite authenticated key agreement protocol using conjugacy problem which works in a braid group. We have proved tha...

To understand what a braid group is, it is easiest to visualize a braid. Consider n strands, all parallel. Consider taking the ith strand and crossing it over the very next strand. This is a braid. In fact, a braid is any sequence of crossings of the bands, provided none of the bands are self-crossing. For instance, a loop, or a band which forms a loop in the middle are not braids. Now, in orde...

We construct an embedding of any right-angled Artin group into a graph braid group. We include an observation which decreases the number of strands of the graph braid group required for this embedding, yielding an example of a hyperbolic surface subgroup embedded in a two strand planar graph braid group.