Actions of the n-strand braid groups on the free group of rank n which are similar to the Artin representation

TL;DR

This paper classifies specific actions of the n-strand braid group on a free group that resemble the Artin representation, and explores related knot invariants.

Contribution

It provides a complete classification of braid group actions on free groups similar to the Artin representation and discusses their implications for knot invariants.

Findings

All such representations are determined and classified.

The paper discusses how these representations relate to knot invariants.

It identifies the structure of actions fixing all but two generators.

Abstract

We classify an action of the $n$-strand braid group on the free group of rank $n$ which is similar to the Artin representation in the sense that the $i$-th generator $\sigma_{i}$ of $B_{n}$ acts so that it fixes all free generators $x_{j}$ except $j = i,i+1$. We determine all such representations and discuss knot invariants coming from such representations.