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Pré-Publication, Document De Travail Année : 2019

Minor theory for surfaces and divides of maximal signature

Résumé

The paper is partially withdrawn: in its current form, Lemma 2.3 is false, so that our proof of Theorem A and Proposition B has an important gap. We were unable to fix it yet. Any help is most welcome. 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.
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Dates et versions

hal-00759428 , version 1 (30-11-2012)
hal-00759428 , version 2 (04-01-2019)

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Sebastian Baader, Pierre Dehornoy. Minor theory for surfaces and divides of maximal signature. 2019. ⟨hal-00759428v2⟩

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