Reidemeister-Turaev torsion of 3-dimensional�Euler structures with simple boundary tangency�and pseudo-Legendrian knots

Abstract

We generalize Turaev's definition of torsion invariants of pairs (M,�), where M is a 3-dimensional manifold andis an Euler structure on M (a non-singular vector field up to homotopy relative to @M and local modifications in Int(M)). Namely, we allow M to have arbitrary boundary andto have simple (convex and/or concave) tangency circles to the boundary. We prove that Turaev's H1(M)-equivariance formula holds also in our generalized context. Our torsions apply in particular to (the exterior of) pseudo- Legendrian knots (i.e. knots transversal to a given vector field), and hence to Legendrian knots in contact 3-manifolds. We show that torsion, as an absolute invariant, contains a lifting to pseudo-Legendrian knots of the classical Alexander invariant. We also precisely analyze the information carried by torsion as a relative invariant of pseudo-Legendrian knots which are framed-isotopic. Using branched standard spines to describe vector fields 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 calculation.