When do links admit homeomorphic C-complexes?

TL;DR

This paper investigates the conditions under which links admit homeomorphic C-complexes, identifying linking numbers as key invariants for 2-component links and Milnor's triple linking number for links with three or more components.

Contribution

It characterizes when two links have homeomorphic C-complexes using linking number invariants, extending the understanding of link equivalences.

Findings

Pairwise linking number fully characterizes 2-component links.

Milnor's triple linking number is the complete obstruction for links with ≥3 components and zero pairwise linking.

Provides criteria for the existence of homeomorphic C-complexes for various link types.

Abstract

Any two knots admit orientation preserving homeomorphic Seifert surfaces, as can be seen by stabilizing. There is a generalization of a Seifert surface to the setting of links called a C-complex. In this paper, we ask when two links will admit orientation preserving homeomorphic C-complexes. In the case of 2-component links, we find that the pairwise linking number provides a complete obstruction. In the case of links with 3 or more components and zero pairwise linking number, Milnor's triple linking number provides a complete obstruction.