Twisted Alexander invariant and non-abelian Reidemeister torsion for hyperbolic three-dimensional manifolds with cusps

Abstract

We study a computational method of the hyperbolic Reidemeister torsion (also called in the literature the non-abelian Reidemeister torsion) induced by J. Porti for complete hyperbolic three-dimensional manifolds with cusps. The derivative of the twisted Alexander invariant for a hyperbolic knot exterior gives the hyperbolic torsion. We prove such a derivative formula of the twisted Alexander invariant for hyperbolic link exteriors like the Whitehead link exterior. We provide the framework for the derivative formula to work, which consists of assumptions on the topology of the manifold and on the representations involved in the definition of the twisted Alexander invariant, and prove derivative formula in that context. We also explore the symmetry properties (with sign) of the twisted Alexander invariant and prove that it is in fact a polynomial invariant, like the usual Alexander polynomial.