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

Valdez-Sánchez, Luis G

doi:10.2140/agt.2025.25.1667

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

Luis G Valdez-Sánchez

Algebraic & Geometric Topology 25 (2025) 1667–1730

DOI: 10.2140/agt.2025.25.1667

Abstract

The Kakimizu complex $MS (K)$ for a knot $K \subset 𝕊^{3}$ 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 $K \subset 𝕊^{3}$ . 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.

Keywords

hyperbolic knot, genus one knot, Seifert torus, Kakimizu complex

Mathematical Subject Classification

Primary: 57K10

Secondary: 57K30

References

Publication

Received: 24 April 2023

Revised: 14 November 2023

Accepted: 10 March 2024

Published: 20 June 2025

Authors

Luis G Valdez-Sánchez
Department of Mathematical Sciences University of Texas at El Paso El Paso, TX United States

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Volume 25, issue 3 (2025)