The crossing matrix and the extended first Johnson homomorphism of a braid group
arXiv:2511.20356
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.
16 pages, 7 figures