[Paper Review] Graph Theoretic Method for Determining non Hurwitz Equivalence in the Braid Group and Symmetric group

Mina Teicher, T. Ben-Itzhak|ArXiv.org|Oct 10, 2001

Geometric and Algebraic Topology1 references3 citations

TL;DR

This paper introduces a graph-theoretic method to determine Hurwitz equivalence of factorizations into transpositions in the symmetric group $S_n$, by associating each factorization with a weighted graph where vertices represent elements and edges represent transpositions. The key result shows that two such factorizations are Hurwitz equivalent if and only if their graphs have identical connected components with matching node sets and edge weights, providing a computationally efficient invariant for distinguishing non-deformation-equivalent surfaces.

ABSTRACT

Motivated by the problem of Hurwitz equivalence of $Δ^2$ factorization in the braid group, we address the problem of Hurwitz equivalence in the symmetric group, obtained by projecting the $Δ^2$ 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.

Motivation & Objective

• To address the problem of Hurwitz equivalence in the symmetric group $S_n$ by analyzing $1_{S_n}$ factorizations into transpositions.
• To provide a computable invariant for distinguishing diffeomorphic but non-deformation-equivalent algebraic surfaces.
• To establish a necessary and sufficient condition for Hurwitz equivalence based on graph structure and component weights.
• To extend results from the braid group $B_n$ to $S_n$ via the natural homomorphism $\phi: B_n \to S_n$, particularly for $\Delta^2$ factorizations.
• To support the computation of the BMT invariant by leveraging graph invariants derived from transposition factorizations.

Proposed method

• Represent each $1_{S_n}$ factorization into transpositions as a weighted graph $G_F = (V_F, E_F)$, where vertices are $\{1, \dots, n\}$ and edges correspond to transpositions.
• Assign weights to edges as the multiplicity of each transposition in the factorization.
• Define the weight of a connected component as the sum of edge weights within it.
• Use Hurwitz moves to transform factorizations into canonical forms by sequentially conjugating adjacent transpositions.
• Apply lemmas (3.9 and 3.10) to show that transposition pairs can be reordered and repositioned within the factorization while preserving equivalence, based on connectivity and shared vertices.
• Prove that any factorization is Hurwitz equivalent to a canonical form determined solely by the number and weights of its connected components.

Experimental results

Research questions

• RQ1 When are two $1_{S_n}$ factorizations into transpositions Hurwitz equivalent?
• RQ2 Can the graph structure of a transposition factorization serve as a complete invariant for Hurwitz equivalence?
• RQ3 How do Hurwitz moves affect the connected components and edge weights of the associated factorization graph?
• RQ4 Can the graph-theoretic structure distinguish between factorizations that are not Hurwitz equivalent, even if they have the same number of factors?
• RQ5 To what extent does the weighted component structure of the graph determine the BMT invariant in algebraic geometry?

Key findings

• Two $1_{S_n}$ factorizations are Hurwitz equivalent if and only if their associated graphs have the same number of connected components, with identical node sets and edge weights across corresponding components.
• The graph structure provides a weaker but computationally efficient invariant for detecting non-deformation-equivalent surfaces that are diffeomorphic.
• Hurwitz equivalence preserves the multiset of connected components with their node sets and total edge weights.
• Any factorization can be transformed via Hurwitz moves into a canonical form determined entirely by the component structure and weights of its graph.
• The proof relies on iterative application of Hurwitz moves to reorder and reweight transpositions, using connectivity and shared vertices to enable reconfiguration.
• The method allows for a systematic reduction of factorizations to a standard form, confirming that equivalence depends only on component-level invariants.