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,&\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 ∂M and modifications supported in a ball contained in Int(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. Using branched standard spines to encode 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. Euler structures of the sort we consider naturally arise in the study of pseudo-Legendrian knots (i.e.~knots transversal to a given vector field), and hence of 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.