The commutator subgroup of the braid group is generated by two elements

TL;DR

This paper determines the minimal number of generators needed for the commutator subgroup of the braid group on n strands, providing explicit generating sets for various values of n.

Contribution

It establishes the smallest possible generating sets for the commutator subgroup of the braid group for specific n, including proving minimality for n=4.

Findings

For n ≥ 7 and n=5, the commutator subgroup is generated by two elements.

For n=4 and n=6, the minimal generating set size is three.

The commutator subgroup of the braid group on 4 strands cannot be generated by fewer than three elements.

Abstract

For $n$ at least 7 and $n$ equal to 5, we give generating sets of size 2 for the commutator subgroup of the braid group on $n$ strands. These generating sets are of the smallest possible cardinality. For $n$ equal to 4 or 6, we give generating sets of size three. We also prove that the commutator subgroup of the braid group on 4 strands cannot be generated by fewer than three elements.