Covering spaces and the Kakimizu complex

TL;DR

This paper proves that the Kakimizu complex, associated with a knot and constructed from minimal genus Seifert surfaces, is simply connected, enhancing understanding of its topological properties.

Contribution

The paper establishes that the Kakimizu complex is simply connected, providing new insights into its topological structure and properties.

Findings

Kakimizu complex is simply connected

Vertices represent isotopy classes of minimal genus Seifert surfaces

Higher simplices correspond to disjoint representatives

Abstract

In 1992, Osamu Kakimizu defined a complex that has become known as the Kakimizu complex of a knot. Vertices correspond to isotopy classes of minimal genus Seifert surfaces of the knot. Higher dimensional simplices correspond to collections of such classes of Seifert surfaces that admit disjoint representatives. We show that this complex is simply connected.