Hodge–type structures as link invariants

Borodzik, Maciej; Némethi, András

Annales de l'Institut Fourier, Tome 63 (2013) no. 1, pp. 269-301.

En se fondant sur des analogies avec la théorie de Hodge des singularités isolées des hypersurfaces, nous construisons des invariants numériques de type de Hodge pour un entrelacs quelconque, pas forcément algébrique, dans une sphère de dimension trois. Nous appelons ces invariants les H-nombres. Ils contiennent la même information sur les entrelacs, que la partie non-dégénérée de la matrice de Seifert modulo S-équivalence réelle. Nous étudions leurs propriétés, en particulier, donnons une formule explicite pour les signatures de Tristram et Levine et les polynômes d’Alexander de haut ordre en termes des H-nombres.

De plus, motivés par la théorie des singularités, nous introduisons le spectre d’un entrelacs, qui, lui aussi, peut être exprimé en termes des H-nombres. Nous établissons quelques propriétés de semicontinuité.

Ces propriétés, peuvent être reliées aux relations skein. Cependant, elles ne sont pas aussi précises que les relations skein classiques.

Based on some analogies with the Hodge theory of isolated hypersurface singularities, we define Hodge–type numerical invariants of any, not necessarily algebraic, link in a three–sphere. We call them H–numbers. They contain the same amount of information as the (non degenerate part of the) real Seifert matrix. We study their basic properties, and we express the Tristram–Levine signatures and the higher order Alexander polynomial in terms of them. Motivated by singularity theory, we also introduce the spectrum of the link (determined from these $H$–numbers), and we establish some semicontinuity properties for it.

These properties can be related with skein–type relations, although they are not so precise as the classical skein relations.

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