# A partial order on multibranched surfaces in 3-manifolds

## TL;DR

This paper introduces a partial order on certain multibranched surfaces within 3-manifolds and characterizes minimal elements among atoroidal and acylindrical cases, advancing understanding of their topological structure.

## Contribution

It defines a new partial order on neighborhood equivalence classes of maximally spread essential multibranched surfaces in 3-manifolds and identifies conditions for minimality.

## Key findings

- Maximally spread essential multibranched surfaces can be partially ordered by neighborhood equivalence.
- Atoroidal and acylindrical surfaces are minimal in this partial order.
- The partial order helps classify multibranched surfaces in 3-manifolds.

## Abstract

In this paper, we introduce a partial order on neighborhood equivalence classes of maximally spread essential multibranched surfaces embedded in a 3-manifold. We show that if a maximally spread essential multibranched surface is atoroidal and acylindrical, then its equivalence class is minimal with respect to the partial order.