Stable equivalence of bridge positions of a handlebody-knot

TL;DR

This paper proves that any two bridge positions of a handlebody-knot can be transformed into each other through a sequence of stabilizations and destabilizations, establishing their stable equivalence.

Contribution

It establishes the stable equivalence of all bridge positions of handlebody-knots, a significant step in understanding their topological properties.

Findings

Any two bridge positions are stably equivalent.

Stable moves connect all bridge positions.

Advances the classification of handlebody-knots.

Abstract

We show that any two bridge positions of a handlebody-knot are stably equivalent.