The density of Lawrence-Krammer and non-conjugate braid representations of links

TL;DR

This paper proves that for generic parameters, the Lawrence-Krammer representation of links is dense in the unitary group, implying infinitely many conjugacy classes of braid representations for most links.

Contribution

It demonstrates the density of the Lawrence-Krammer representation for generic parameters using Lie group theory and unitarization, revealing new properties of link representations.

Findings

Representation is dense in the unitary group for generic parameters.

Most links have infinitely many conjugacy classes of braid representations.

Density holds for certain subgroups and non-minimal strand numbers.

Abstract

We use some Lie group theory and Budney's unitarization of the Lawrence-Krammer representation, to prove that for generic parameters of definite form the image of the representation (also on certain types of subgroups) is dense in the unitary group. This implies that, except possibly for closures of full-twist braids, all links have infinitely many conjugacy classes of braid representations on any non-minimal number of (and at least 4) strands.