Reidemeister Torsion of 3-Dimensional Euler Structures with Simple Boundary Tangency and Legendrian Knots

Abstract

We generalize Turaev's definition of torsion invariants of pairs $(M,\xi)$, where $M$ is a 3-dimensional manifold and $\xi$ is an Euler structure on $M$ (a non-singular vector field up to homotopy relative to the boundary of $M$ and local modifications in the interior of $M$). Namely, we allow $M$ to have arbitrary boundary and $\xi$ to have simple (convex and/or concave) tangency circles to the boundary. We prove that Turaev's $H_1(M)$-equivariance formula holds also in our generalized context. Our torsions apply in particular to (the exterior of) Legendrian links (in particular, knots) in contact 3-manifolds, and we prove that they can distinguish knots which are isotopic as framed knots but not as Legendrian knots. Using the combinatorial encoding of vector fields based on branched standard spines we show how to explicitly invert Turaev's reconstruction map from combinatorial to smooth Euler structures, thus making the computation of torsions a more effective one. As an example we work out a specific computation.