A locally minimal, but not globally minimal bridge position of a knot

TL;DR

This paper presents an example of a knot bridge position that is locally minimal but not globally minimal, demonstrating that bridge positions cannot always be simplified to the minimal bridge number.

Contribution

It introduces a specific example of an unstabilized, nonminimal bridge position, highlighting limitations in the simplification process of knot bridge positions.

Findings

Existence of a locally minimal but not globally minimal bridge position

Bridge position simplification does not always reduce bridge number monotonically

Challenges assumptions about minimality in knot bridge positions

Abstract

We give a locally minimal, but not globally minimal bridge position of a knot, that is, an unstabilized, nonminimal bridge position of a knot. It implies that a bridge position cannot always be simplified so that the bridge number monotonically decreases to the minimal.