On links with locally infinite {K}akimizu complexes

TL;DR

This paper demonstrates that the Kakimizu complex of certain knots can be locally infinite and characterizes the links with this property as satellites of specific knot types, answering a previously open question.

Contribution

It shows the existence of locally infinite Kakimizu complexes for knots and classifies links with this property as satellites of torus, cable, or connected sum knots.

Findings

Kakimizu complex of some knots can be locally infinite

Links with locally infinite Kakimizu complexes are satellites of specific knots

Characterization of such links as satellites of torus, cable, or connected sums

Abstract

We show that the Kakimizu complex of a knot may be locally infinite, answering a question of Przytycki--Schultens. We then prove that if a link $L$ only has connected Seifert surfaces and has a locally infinite Kakimizu complex then $L$ is a satellite of either a torus knot, a cable knot or a connected sum, with winding number 0.