Graph Theoretic Method for Determining non Hurwitz Equivalence in the Braid Group and Symmetric group

TL;DR

This paper introduces a graph-theoretic approach to determine Hurwitz equivalence of factorizations in the symmetric group, facilitating the classification of surface diffeomorphisms and computing invariants.

Contribution

It establishes a novel graph-based criterion for Hurwitz equivalence in the symmetric group derived from braid group factorizations, simplifying the analysis of surface invariants.

Findings

Hurwitz equivalence corresponds to identical weighted graph components.

Graph structure provides an easily computable invariant for surface classification.

Main result aids in computing the Braid Monodromy Type invariant.

Abstract

Motivated by the problem of Hurwitz equivalence of $\Delta ^2$ factorization in the braid group, we address the problem of Hurwitz equivalence in the symmetric group, obtained by projecting the $\Delta ^2$ factorizations into $S_n$. We get $1_{S_n}$ factorizations with transposition factors. Looking at the transpositions as the edges in a graph, we show that two factorizations are Hurwitz equivalent if and only if their graphs have the same weighted connected components. The main result of this paper will help us to compute the "Braid Monodromy Type" invariant. The graph structure gives a weaker but very easy to compute invariant to distinguish between diffeomorphic surfaces which are not deformation of each other.