Non–abelian Reidemeister Torsion for Twist Knots

Twist knots form a family of special two–bridge knots which include the trefoil knot and the figure eight knot. The knot group of a two–bridge knot has a particularly nice presentation with only two generators and a single relation. One could find our interest in this family of knots in the following facts: first, twist knots except the trefoil knot are hyperbolic; and second, twist knots are not fibered except the trefoil knot and the figure eight knot (see Remark 2 of the present paper for details). The non–abelian Reidemeister torsion associated to a representation of a knot group to a general linear group over a field has been studied since the early 1990’s. It was initially considered as a twisted Alexander polynomial by Lin [Lin01], Wada [Wad94], and later interpreted as a form of Reidemeister torsion by Kitano [Kit96], Kirk-Livingston [KL99] and Goda–Kitano and Morifuji [GKM05]. This invariant in many cases is stronger than the classical ones, for example it detects the unknot [SW06], and decides fiberness for knots of genus one [FV07]. Abelian Reidemeister torsions are now well–known, see e.g. Turaev’s monograph [Tur02]; but unfortunately in the case of non–abelian representations concrete computations of such torsions are still very few. In [Por97], Porti began the study of the non–abelian Reidemeister torsion (consider as a functional on the non–abelian part of the character variety) with the adjoint representation associated to an irreducible representation of the fundamental group of a hyperbolic three–dimensional manifold to SL2(C). In [Dub05], the first author introduced a sign–refined version of this torsion. In the present paper, we call this (sign–refined) torsion the SL2(C)−non–abelian Reidemeister torsion. One can observe that this torsion has connections with hyperbolic structures, the theory of character variety, and the theory of Chern-Simons invariant, see e.g. [DK07]. In [Dub06, Main Theorem], one can find an “explicit” formula which gives the value of the non–abelian Reidemeister torsion for fibered knots in terms of the map induced by the monodromy of the knot at the level of the character variety of the knot exterior. In particular, a practical formula of the non–abelian Reidemeister torsion for torus knots is presented in [Dub06, Section 6.2]. One can also find an explicit formula for the non–abelian Reidemeister torsion for the figure eight knot in [Dub06, Section 7]. More recently, the last author found [Yam05, Theorem 3.1.2] an interpretation of the non–abelian Reidemeister torsion in terms of a sort of twisted Alexander polynomial (called in this paper, the non– abelian Reidemeister torsion polynomial) and gave an explicit formula of the non–abelian torsion for the twist knot 52.