A Legendrian Turaev torsion via generating families

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.