Characterization of 3-bridge links with infinitely many 3-bridge spheres

TL;DR

This paper characterizes prime, unsplittable 3-bridge links in S^3 that admit infinitely many 3-bridge spheres, showing they belong to a specific infinite family previously constructed.

Contribution

It proves that only links from a particular family admit infinitely many 3-bridge spheres, providing a complete classification.

Findings

Prime, unsplittable 3-bridge links with infinitely many 3-bridge spheres are exactly those in the constructed family.

The paper establishes a characterization theorem for such links.

It confirms the exclusivity of the family for links with infinitely many 3-bridge spheres.

Abstract

The author, in her previous paper, constructed an infinite family of 3-bridge links each of which admits infinitely many 3-bridge spheres up to isotopy. In this paper, we prove that if a prime, unsplittable link $L$ in $S^3$ admits infinitely many 3-bridge spheres up to isotopy then $L$ belongs to the family.