Matrix transposition and braid reversion

TL;DR

This paper explores how matrix transposition affects the classification of semi-simple braid group representations, identifying fixed points and their role in detecting braid-reversion.

Contribution

It provides a classification of fixed-point components under matrix transposition and links these to braid-reversion detection in representation varieties.

Findings

Connected components can detect braid-reversion.

Fixed-point components are classified.

Involution acts as identity on certain components.

Abstract

Matrix transposition induces an involution on the isomorphism classes of semi-simple n-dimensional representations of the three string braid group. We show that a connected component of this variety can detect braid-reversion or that the involution acts as the identity on it. We classify the fixed-point components.