Denseness and Zariski denseness of Jones braid representations

TL;DR

This paper proves that Jones braid group representations are dense in the complex Zariski topology for most parameters, using representation and group theory tools without relying on classification theorems.

Contribution

It establishes the Zariski denseness of Jones representations for non-root of unity parameters and computes their real Zariski closure, extending previous results.

Findings

Jones representations are dense in the complex Zariski topology for non-root of unity parameters.

The real Zariski closure of these representations is computed.

When indiscrete in the analytic topology, the analytic and real Zariski closures coincide.

Abstract

Using various tools from representation theory and group theory, but without using hard classification theorems such as the classification of finite simple groups, we show that the Jones representations of braid groups are dense in the complex Zariski topology when the parameter $t$ is not a root of unity. As first established by Freedman, Larsen, and Wang, we the same result when t is a non-lattice root of unity, other than one initial case when t has order 10. We also compute the real Zariski closure of these representations. When such a representation is indiscrete in the analytic topology, then its analytic closure is the same as its real Zariski closure.