Genus one hyperbolic knots in the 3-sphere and the Kakimizu complex

Luis G. Valdez-Sanchez (UTEP)

Abstract

The Kakimizu complex $MS(K)$ for a knot $K\subset\mathbb{S}^3$ is the simplicial complex with simplices the collections of isotopy classes of minimal genus Seifert surfaces in the knot exterior that can be realized by mutually disjoint and non-parallel such surfaces.

For genus one hyperbolic knots $K\subset\mathbb{S}^3$ the optimal bounds for the dimension and diameter of the complex $MS(K)$ are known to be $4$ and $2$, respectively. We refine these results by showing that, in the top dimension, the complex ${MS}(K)$ consists of at most $2$ simplices, and exactly one $4$-simplex in the $4$-dimensional case. We also provide infinitely many examples of such knots for which $MS(K)$ consists of exactly one or exactly two top-dimensional simplices.

Date

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