Topological complexity of subgroups of Artin's braid groups

TL;DR

This paper investigates the topological complexity of specific subgroups of Artin's braid groups, providing bounds and extending known results to more general settings including higher topological complexity.

Contribution

It introduces bounds for the topological complexity of subgroups of braid groups fixing strands and generalizes results to higher topological complexity.

Findings

Topological complexity of subgroups fixing two strands is 2n-3.

Provides upper and lower bounds for mixed braid groups.

Extends results to higher topological complexity.

Abstract

We consider the topological complexity of subgroups of Artin's braid group consisting of braids whose associated permutations lie in some specified subgroup of the symmetric group. We give upper and lower bounds for the topological complexity of such mixed braid groups. In particular we show that the topological complexity of any subgroup of the n-strand braid group which fixes any two strands is 2n-3, extending a result of Farber and Yuzvinsky in the pure braid case. In addition, we generalise our results to the setting of higher topological complexity.