Crossing matrix and a polynomial invariant of braid systems up to Hurwitz equivalence

TL;DR

This paper introduces a new polynomial invariant for braid systems based on crossing matrices, invariant under Hurwitz equivalence, aiding the analysis of surface braids and links.

Contribution

It presents a novel polynomial invariant for braid systems and applies it to surface braids and links, providing tools for understanding their transformations.

Findings

Defined a crossing matrix for braids

Introduced a polynomial invariant invariant under Hurwitz moves

Provided an indicator for Euler fusion or fission in surface braids

Abstract

We study the crossing matrix of a braid and introduce a polynomial invariant for braid systems that is invariant under Hurwitz equivalence. As an application to the study of surface braids and surface links, we also define an invariant that can be used as an indicator of the necessity of Euler fusion or fission between braid systems.