The Kakimizu complex of a surface

TL;DR

This paper extends the concept of the Kakimizu complex from knots to surfaces and 3-manifolds, revealing new topological properties such as contractibility and non-quasi-Euclidean behavior.

Contribution

It generalizes the Kakimizu complex to surfaces and 3-manifolds and demonstrates their contractibility and potential non-quasi-Euclidean nature.

Findings

Kakimizu complexes of surfaces are contractible.

They can be non-quasi-Euclidean.

Existence of 3-manifolds with non-quasi-Euclidean Kakimizu complexes.

Abstract

The Kakimizu complex is usually defined in the context of knots, where it is known to be quasi-Euclidean. We here generalize the definition of the Kakimizu complex to surfaces and 3-manifolds (with or without boundary). Interestingly, in the setting of surfaces, the complexes and the techniques turn out to replicate those used to study the Torelli group, {\it i.e.,} the "nonlinear" subgroup of the mapping class group. Our main results are that the Kakimizu complexes of a surface are contractible and that they need not be quasi-Euclidean. It follows that there exist (product) $3$-manifolds whose Kakimizu complexes are not quasi-Euclidean.