Rationality of the SL(2,C)-Reidemeister torsion in dimension 3

Abstract

If $M$ is a finite volume complete hyperbolic 3-manifold with one cusp and no 2-torsion, the geometric component $X_M$ of its $\SL(2,\BC)$-character variety is an affine complex curve, which is smooth at the discrete faithful representation ${\rho }_0$. Porti defined a non-abelian Reidemeister torsion in a neighborhood of ${\rho }_0$ in $X_M$ and observed that it is an analytic map, which is the germ of a unique rational function on $X_M$. In the present paper we prove that (a) the torsion of a representation lies in at most quadratic extension of the invariant trace field of the representation, and (b) the existence of a polynomial relation of the torsion of a representation and the trace of the meridian or the longitude. We postulate that the coefficients of the $1/N^k$-asymptotics of the Parametrized Volume Conjecture for $M$ are elements of the field of rational functions on $X_M$.

13 pages

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• On the characteristic and deformation varieties of a knot
• Asymptotics of the colored Jones function of a knot
• SL(2,C) Chern-Simons theory and the asymptotic behavior of the colored Jones polynomial
• Complex valued Ray-Singer torsion
• Non-triviality of the A-polynomial for knots in S^3
• Chern-Simons theory, analytic continuation and arithmetic
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Cited by in corpus (1)

• Reidemeister torsion, hyperbolic three-manifolds, and character varieties