The Kakimizu complex for genus one hyperbolicknots in the 3-sphere

The Kakimizu complex MS.K/ for a knot K 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.We determine the structure of the Kakimizu complex MS.K/ of genus one hyperbolic knots K 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 D 0; 4 and otherwise of at most two d -simplices which intersect in a common .d1/-face.For the cases 1 d 3 we also construct infinitely many examples of such knots where MS.K/ consists of two d -simplices.