The Kakimizu complex for genus one hyperbolic knots in the 3-sphere

Abstract

The Kakimizu complex $MS(K)$ for a knot $K\subset\mathbb{S}^3$ is the simplicial complex with vertices the isotopy classes of minimal genus Seifert surfaces in the exterior of $K$ and simplices any set of vertices with mutually disjoint representative surfaces. In this paper we determine the structure of the Kakimizu complex $MS(K)$ of genus one hyperbolic knots $K\subset\mathbb{S}^3$. In contrast with the case of hyperbolic knots of higher genus, it is known that the dimension $d$ of $MS(K)$ is universally bounded by $4$, and we show that $MS(K)$ consists of a single $d$-simplex for $d=0,4$ and otherwise of at most two $d$-simplices which intersect in a common $(d-1)$-face. For the cases $1\leq d\leq 3$ we also construct infinitely many examples of such knots where $MS(K)$ consists of two $d$-simplices.