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

Valdez-Sánchez, Luis G¹

¹Department of Mathematical Sciences, University of Texas at El Paso, El Paso, TX, United States

Algebraic and Geometric Topology, Tome 25 (2025) no. 3, pp. 1667-1730

Résumé

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 (d−1)-face. For the cases 1 ≤ d ≤ 3 we also construct infinitely many examples of such knots where MS ⁡ (K) consists of two d-simplices.

Détail

DOI : 10.2140/agt.2025.25.1667

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

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