A Linear Bound on the Diameter of the Kakimizu Complex for Hyperbolic Knots

TL;DR

This paper establishes a linear upper bound on the diameter of the Kakimizu complex for hyperbolic knots, confirming a conjecture and providing new insights into the structure of incompressible Seifert surfaces.

Contribution

It introduces a new complex $IS_ ext{ell}(K)$ and proves its diameter is linearly bounded, leading to a linear bound on the Kakimizu complex's diameter for hyperbolic knots.

Findings

The complex $IS_ ext{ell}(K)$ is connected.

The diameter of $IS_ ext{ell}(K)$ has a linear upper bound in $ ext{ell}$.

The diameter of the Kakimizu complex $MS(K)$ grows linearly with the genus $g$, bounded above by $6g-4$.

Abstract

This paper focuses on the Kakimizu complex of a hyperbolic knot $K$. We define a complex $IS_\ell(K)$ to study incompressible Seifert surfaces of genus at most $\ell$, and prove that it is connected and that its diameter admits a linear upper bound in terms of $\ell$. As a corollary, we show that the diameter of the Kakimizu complex $MS(K)$ of a hyperbolic knot grows linearly with the genus $g$, confirming a conjecture of Sakuma--Shackleton. More precisely, it is bounded above by $6g-4$.