Rationality and dense families of B(3) representations

TL;DR

This paper studies the structure of B(3)-representations, showing dense families of simple representations in dimensions under 12, and identifies a unique component detecting braid-reversion, with explicit parametrizations.

Contribution

It provides explicit rational parametrizations of dense simple B(3)-representations in all dimensions less than 12 and identifies a unique component detecting braid-reversion.

Findings

Existence of dense families of simple B(3)-representations in all dimensions < 12

Identification of a unique component detecting braid-reversion

Explicit rational parametrizations of these representations

Abstract

Every irreducible component of the variety of semi-simple n-dimensional representations of the modular group has a Zariski dense subset contained in the image of an etale map from a rational quotient variety of representations of a fixed quiver Q. As an application we prove that there is a unique component of 6-dimensional representations of the three string braid group B(3) able to detect braid-reversion. Further, we give explicit rational parametrizations of dense families of simple B(3)-representations in all dimensions < 12.