Reidemeister Torsion of 3Dimensional Euler Structures with Simple Boundary Tangency and PseudoLegendrian Knots
Abstract
We generalize Turaev's definition of torsion invariants of pairs (M,x), where M is a 3dimensional manifold and x is an Euler structure on M (a nonsingular 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) pseudoLegendrian knots (i.e. knots transversal to a given vector field), and hence to Legendrian knots in contact 3manifolds. We show that torsion, as an absolute invariant, contains a lifting to pseudoLegendrian knots of the classical Alexander invariant. We also precisely analyze the information carried by torsion as a relative invariant of pseudoLegendrian knots which are framedisotopic. 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 February 2000
 arXiv:
 arXiv:math/0002143
 Bibcode:
 2000math......2143B
 Keywords:

 Mathematics  Geometric Topology;
 57M27
 EPrint:
 This is a revised version of math.GT/9907184, submitted as a new paper because the understanding of torsion of pseudoLegendrian knots is here substantially improved