The crossing matrix and the extended first Johnson homomorphism of a braid group

TL;DR

This paper compares two different crossed homomorphisms on a braid group, demonstrating their equivalence and providing detailed computations for simple braids, enhancing understanding of algebraic and diagrammatic approaches.

Contribution

It establishes the equivalence of diagrammatic and algebraic crossed homomorphisms on braid groups and computes them explicitly for basic braid elements.

Findings

The two crossed homomorphisms are essentially the same.

Explicit formulas are provided for simple braids.

The results unify algebraic and diagrammatic perspectives.

Abstract

We compare two crossed homomorphisms on a braid group, one defined diagrammatically and the other defined algebraically. We show that these crossed homomorphisms are essentially the same, and compute them in detail for simple braids, namely elements conjugate to the standard generators of the braid group or to their inverses.