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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
Abstract

The Kakimizu complex MS (K) for a knot K 𝕊3 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 𝕊3. 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 (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.

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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