Neighborhood equivalence for multibranched surfaces in 3-manifolds

TL;DR

This paper introduces moves for multibranched surfaces in 3-manifolds that enable transforming any two surfaces with the same neighborhood into each other through finitely many steps, aiding in their classification.

Contribution

It presents a set of moves for multibranched surfaces in 3-manifolds that establish neighborhood equivalence, providing a new method for their analysis.

Findings

Moves connect any two multibranched surfaces with the same neighborhood

Finiteness of steps ensures practical applicability

Advances understanding of multibranched surface classification

Abstract

A multibranched surface is a 2-dimensional polyhedron without vertices. We introduce moves for multibranched surfaces embedded in a 3-manifold, which connect any two multibranched surfaces with the same regular neighborhoods in finitely many steps.