A Legendrian Turaev torsion via generating families

Álvarez-Gavela, Daniel; Igusa, Kiyoshi

Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 57-119.

Nous introduisons un invariant des sous-variétés legendriennes construit à l’aide de fonctions génératrices. Cet invariant est défini pour une certaine classe de sous-variétés legendriennes, que nous appelons de type d’Euler, dans un espace de $1$-jets. Nous utilisons cet invariant pour étudier les mailles legendriennes : une famille de sous-variétés legendriennes de type d’Euler dont le motif d’entrelac est déterminé par un graphe bicolore et trivalent qui est muni d’un ordre cyclique des arêtes concourantes en un même sommet. La torsion de Turaev d’une maille legendrienne est reliée à une certaine monodromie de glissement d’anses, que nous calculons en terme de la combinatoire du graphe. Comme application, nous exhibons, dans l’espace des $1$-jets de toute surface fermée orientable, des paires d’entrelacs legendriens qui sont formellement équivalents, ne peuvent être distingués par aucun invariant legendrien naturel, et pourtant ne sont pas isotopes parmi les variétés legendriennes. Ces exemples sont apparus sous une forme différente dans les travaux du second auteur avec J. Klein sur des dessins pour $K_3$ et sur la torsion de Reidemeister supérieure de fibrés en cercles.

We introduce a Legendrian invariant built out of the Turaev torsion of generating families. This invariant is defined for a certain class of Legendrian submanifolds of $1$-jet spaces, which we call of Euler type. We use our invariant to study mesh Legendrians: a family of $2$-dimensional Euler type Legendrian links whose linking pattern is determined by a bicolored trivalent ribbon graph. The Turaev torsion of mesh Legendrians is related to a certain monodromy of handle slides, which we compute in terms of the combinatorics of the graph. As an application, we exhibit pairs of Legendrian links in the $1$-jet space of any orientable closed surface which are formally equivalent, cannot be distinguished by any natural Legendrian invariant, yet are not Legendrian isotopic. These examples appeared in a different guise in the work of the second author with J. Klein on pictures for $K_3$ and the higher Reidemeister torsion of circle bundles.

Classification : 57R17, 19J10
Keywords: Legendrians, Turaev torsion, K-theory
Mot clés : Legendriennes, torsion de Turaev, K-théorie

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