An Infinite Family of Links with Critical Bridge Spheres

TL;DR

This paper constructs the first known examples of critical bridge spheres for nontrivial links in 3-manifolds, advancing the understanding of minimal surfaces in knot theory.

Contribution

It introduces a combinatorial approach to construct critical bridge spheres for nontrivial links, providing new examples in the study of topologically minimal surfaces.

Findings

First known critical bridge spheres for nontrivial links

Uses combinatorial definition of critical surfaces

Advances understanding of minimal surfaces in link complements

Abstract

A closed, orientable, splitting surface in an oriented $3$-manifold is a topologically minimal surface of index $n$ if its associated disk complex is $(n-2)$-connected but not $(n-1)$-connected. A critical surface is a topologically minimal surface of index $2$. In this paper, we use an equivalent combinatorial definition of critical surfaces to construct the first known critical bridge spheres for nontrivial links.