Bridge Positions of Links That Cannot Be Monotonically Simplified

TL;DR

This paper constructs infinite families of links with specific bridge positions, demonstrating that certain bridge positions cannot be simplified monotonically, highlighting complex behaviors in link theory.

Contribution

It introduces a method to produce links with both locally minimal and globally minimal bridge positions for different bridge numbers, using Takao et al.'s criterion.

Findings

Existence of links with unperturbed bridge spheres at local minima

Demonstration of non-monotonic bridge position simplification

Infinite families of such links for given integers m and n

Abstract

For any pair of integers $m$ and $n$ such that $3<m<n$, we provide an infinite family of links, where each link in the family has a locally minimal $n$-bridge position and a globally minimal $m$-bridge position. We accomplish this by applying the criterion of Takao et al. The $n$-bridge position is interesting because the corresponding bridge sphere is unperturbed, so it must be perturbed at least once before it can be de-perturbed to attain a globally minimal $m$-bridge sphere.