Braids, their Properties and Generalizations

For a general connected surface M and an arbitrary braid α from the surface braid group B n−1 (M ), we study the system of equations d 1 β = . . . = d n β = α, where the operation d i is the removal of the ith strand. We prove that for M = S 2 and M = RP 2 , this system of equations has a solution β ∈ B n (M ) if and only if d 1 α = . . . = d n−1 α. We call the set of braids satisfying the last system of equations Cohen braids. We study Cohen braids and prove that they form a subgroup. We also construct a set of generators for the group of Cohen braids. In the cases of the sphere and the projective plane we give some examples for a small number of strands.

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“…General properties. Recall that the generators for the pure braid group P n are given by [10], for example).…”

Section: Cohen and Brunnian Braids In The Braid Group Of The Diskmentioning

confidence: 99%

On Cohen braids

Bardakov

Vershinin²,

Wu³

2014

Proc. Steklov Inst. Math.

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“…Many of these generalized braid groups are of importance in their own rights, e.g. [Bel04], [BG12], [Dam17], [Ver06], [Ver14]. In this paper, we consider two such families of generalized braid groups and investigate their commutator subgroups.…”

Section: Introductionmentioning

confidence: 99%

Commutator subgroups of welded braid groups

Dey

Gongopadhyay

2018

Topology and its Applications

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The singular braids with n strands, n ≥ 3, were introduced independently by Baez and Birman. It is known that the monoid formed by the singular braids is embedded in a group that is known as singular braid group, denoted by SGn. There has been another generalization of braid groups, denoted by GV Bn, n ≥ 3, which was introduced by Fang as a group of symmetries behind quantum quasi-shuffle structures. The group GV Bn simultaneously generalizes the classical braid group, as well as the virtual braid group on n strands.We investigate the commutator subgroups SG ′ n and GV B ′ n of these generalized braid groups. We prove that SG ′ n is finitely generated if and only if n ≥ 5, and GV B ′ n is finitely generated if and only if n ≥ 4. Further, we show that both SG ′ n and GV B ′ n are perfect if and only if n ≥ 5.

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“…The pure braid group P n (of a disc) can be given by the following presentation: generators: a i,j , 1 ≤ i < j ≤ n, the defining relations (Burau relations ([5], [20])):…”

Section: Introductionmentioning

confidence: 99%