Minor theory for surfaces and divides of maximal signature

Abstract

We prove that the restriction of surface minority to fiber surfaces of divides is a wellquasi-order. Here surface minority is the partial order on isotopy classes of surfaces embedded in R 3 associated with incompressible subsurfaces. The proof relies on a refinement of the Robertson-Seymour Theorem that involves colored graphs embedded into the disc. 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.

Well-quasi-order of plane minors and an application to link diagrams

“…A further application to knot theory. As in [5], we became interested in the plane minor relation (in our case, in the sphere minor relation) coming from a problem in knot theory.…”

“…Besides its independent interest, this result is relevant in view of its applications to knot theory. Plane minors are related to linking graphs associated to positive braids [6] (see also [4,40]), and to the surface minor relation on embedded surfaces in R 3 [5].…”