Contractibility of the Kakimizu complex and symmetric Seifert surfaces

TL;DR

This paper proves the contractibility of the Kakimizu complex and its generalizations, establishing the existence of symmetric Seifert surfaces through fixed-point set analysis in the context of knot theory.

Contribution

It demonstrates the contractibility of the Kakimizu complex and its generalizations, confirming a conjecture and linking fixed-point sets to symmetric Seifert surfaces.

Findings

Kakimizu complex is contractible.

Fixed-point sets for subgroups are contractible or empty.

Existence of symmetric Seifert surfaces for finite subgroups.

Abstract

Kakimizu complex of a knot is a flag simplicial complex whose vertices correspond to minimal genus Seifert surfaces and edges to disjoint pairs of such surfaces. We discuss a general setting in which one can define a similar complex. We prove that this complex is contractible, which was conjectured by Kakimizu. More generally, the fixed-point set (in the Kakimizu complex) for any subgroup of an appropriate mapping class group is contractible or empty. Moreover, we prove that this fixed-point set is non-empty for finite subgroups, which implies the existence of symmetric Seifert surfaces.