Braid Groups are Linear

TL;DR

This paper proves that the Lawrence-Krammer representation provides a faithful linear representation for all braid groups B_n, confirming their linearity and advancing understanding of their algebraic structure.

Contribution

It extends the faithfulness of the Lawrence-Krammer representation from n=4 to all n, establishing the linearity of all braid groups.

Findings

Lawrence-Krammer representation is faithful for all n

Braid groups are linear for all n

Advances understanding of braid group algebraic properties

Abstract

The braid groups B_n can be defined as the mapping class group of the n-punctured disc. The Lawrence-Krammer representation of the braid group B_n is the induced action on a certain twisted second homology of the space of unordered pairs of points in the n-punctured disc. Recently, Daan Krammer showed that this is a faithful representation in the case n=4. In this paper, we show that it is faithful for all n.