The coarse geometry of the Kakimizu complex

TL;DR

This paper demonstrates that the Kakimizu complex associated with minimal genus Seifert surfaces of a knot in the 3-sphere has a geometric structure similar to a Euclidean lattice, revealing its coarse geometric properties.

Contribution

It establishes that the Kakimizu complex is quasi-isometric to a Euclidean lattice, providing new insights into its large-scale geometric structure.

Findings

Kakimizu complex is quasi-isometric to  lattice

Provides a geometric classification of the Kakimizu complex

Advances understanding of the topology of knot complements

Abstract

We show that the Kakimizu complex of minimal genus Seifert surfaces for a knot in the 3-sphere is quasi-isometric to a Euclidean integer lattice $\mathbb Z^n$ for some $n \geq 0$.