A Legendrian Turaev torsion via generating families

TL;DR

This paper introduces a new Legendrian invariant based on Turaev torsion for specific Legendrian submanifolds, enabling the distinction of Legendrian links that are otherwise indistinguishable by existing invariants.

Contribution

It defines a novel Legendrian invariant using Turaev torsion for Euler type Legendrians and applies it to distinguish Legendrian links with identical classical invariants.

Findings

The invariant is computable for mesh Legendrians using graph combinatorics.

It can distinguish Legendrian links that are formally equivalent but not Legendrian isotopic.

Examples demonstrate the invariant's effectiveness in complex Legendrian link classification.

Abstract

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.