Hopf Arborescent Links, Minor Theory, and Decidability of the Genus Defect

Pierre Dehornoy (1) , Corentin Lunel (2) , Arnaud de Mesmay (2)

1 I2M - Institut de Mathématiques de Marseille
2 LIGM - Laboratoire d'Informatique Gaspard-Monge

While the problem of computing the genus of a knot is now fairly well understood, no algorithm is known for its four-dimensional variants, both in the smooth and in the topological locally flat category. In this article, we investigate a class of knots and links called Hopf arborescent links, which are obtained as the boundaries of some iterated plumbings of Hopf bands. We show that for such links, computing the genus defects, which measure how much the four-dimensional genera differ from the classical genus, is decidable. Our proof is non-constructive, and is obtained by proving that Seifert surfaces of Hopf arborescent links under a relation of minors defined by containment of their Seifert surfaces form a well-quasi-order.

Mots clés: Well-quasi-order; Genus; 4-genus; Arborescent links

Domaines: Topologie géométrique [math.GT]

Pierre Dehornoy, Corentin Lunel, Arnaud de Mesmay. Hopf Arborescent Links, Minor Theory, and Decidability of the Genus Defect. 2024. ⟨hal-04552398⟩