Bridge surfaces with the topological minimality preserved by perturbation

TL;DR

This paper demonstrates that for knots with bridge surfaces that are topologically minimal, perturbations preserve their minimality and increase the index by at most one, except when n=2.

Contribution

It establishes a relationship between perturbations and the topological minimality index of bridge surfaces for knots, extending understanding of their stability.

Findings

Perturbations increase the minimality index by at most one.

Topologically minimal bridge surfaces are stable under perturbation for all n ≠ 2.

The minimality property is preserved with a controlled increase in index.

Abstract

We show that except for $n = 2$ if a bridge surface for a knot is an index $n$ topologically minimal surface, then after a perturbation it is still topologically minimal with index at most $n+1$.