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

This paper focuses on the Kakimizu complex of a hyperbolic knot $K$. We define a complex $I{S}_{\mathrm{\ell }}(K)$ to study incompressible Seifert surfaces of genus at most $\mathrm{\ell }$, and prove that it is connected and that its diameter admits a linear upper bound in terms of $\mathrm{\ell }$. 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 $6g-4$.