Minor theory for surfaces and divides of maximal signature

Sebastian Baader (1) , Pierre Dehornoy (1)

1 Mathematisches Institut [Bern]

We prove that the restriction of surface minority to fiber surfaces of divides is a well-quasi-order. Here surface minority is the partial order on isotopy classes of surfaces embedded in the 3-space associated with incompressible subsurfaces. The proof relies on a refinement of the Robertson-Seymour Theorem that involves colored graphs embedded into the plane. Our result implies that every property of fiber surfaces of divides that is preserved by surface minority is characterized by a finite number of prohibited minors. For the signature to be equal to the first Betti number is such a property. We explicitly determine the corresponding prohibited minors. As an application we establish a correspondance between divide links of maximal signature and Dynkin diagrams.

Mots clés: graph minor; embedded surface; incompressible surface; knot; signature; divide

Domaines: Topologie géométrique [math.GT]; Combinatoire [math.CO]

Sebastian Baader, Pierre Dehornoy. Minor theory for surfaces and divides of maximal signature. 2012. ⟨hal-00759428v1⟩