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

TL;DR

This paper extends Turaev's torsion invariants to 3-manifolds with boundary and simple boundary tangency, applying to pseudo-Legendrian knots and providing explicit computational methods.

Contribution

It generalizes torsion invariants to manifolds with boundary and simple tangency, and connects these to pseudo-Legendrian knots and their classical invariants.

Findings

Torsion invariants are extended to manifolds with boundary and simple tangency.

Torsion contains a lift of the classical Alexander invariant for pseudo-Legendrian knots.

Explicit methods for computing torsions using branched spines are developed.

Abstract

We generalize Turaev's definition of torsion invariants of pairs (M,x), where M is a 3-dimensional manifold and x is an Euler structure on M (a non-singular vector field up to homotopy relative to bM and local modifications in int(M). Namely, we allow M to have arbitrary boundary and x 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) 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.