A knot with destabilized bridge spheres of arbitrarily high bridge number

TL;DR

This paper constructs infinite families of knots with destabilized bridge spheres of arbitrarily high bridge number, demonstrating complex bridge sphere structures in knot theory.

Contribution

It introduces explicit examples of knots with destabilized bridge spheres across a wide range of bridge numbers, expanding understanding of bridge sphere destabilization.

Findings

Existence of knots with destabilized (2k+5)-bridge spheres for all k≥0

Existence of knots with both 3-bridge and n-bridge destabilized spheres for n≥4

Demonstrates the complexity of bridge sphere destabilization in knot theory

Abstract

We show that there exists an infinite family of knots, each of which has, for each integer k>=0, a destabilized (2k+5)-bridge sphere. We also show that, for each integer n>=4, there exists a knot with a destabilized 3-bridge sphere and a destabilized n-bridge sphere.