No abstract

Abstract. We show a relationship between the non-acyclic Reidemeister torsion and a zero of the acyclic Reidemeister torsion for a λ-regular SU(2) or SL(2, C)-representation of a knot group. Then we give a method to calculate the non-acyclic Reidemeister torsion of a knot exterior. We calculate a new example and investigate the behavior of the non-acyclic Reidemeister torsion associated to a 2-bridge knot and SU(2)-representations of its knot group.

Let K be a knot in an integral homology 3-sphere Σ with exterior E K , and let B 2 denote the 2-fold branched cover of Σ branched along K. We construct a map Φ from the slice of characters with trace free along meridians in the SL 2 (C)-character variety of the knot exterior E K to the SL 2 (C)-character variety of 2-fold branched cover B 2 . When this map is surjective, it describes the slice as the 2-fold branched cover over the SL 2 (C)-character variety of B 2 with branched locus given by the abelian characters, whose preimage is precisely the set of metabelian characters. We show that each of metabelian character can be represented as the character of a binary dihedral representation of π 1 (E K ). The map Φ is shown to be surjective for all 2-bridge knots and all pretzel knots of type (p, q, r). An extension of this framework to n-fold branched covers is also described.1991 Mathematics Subject Classification. 57M27 and 57M05 and 57M12. Key words and phrases. Character varieties and branched covers and binary dihedral representations and knots and metabelian representations. 1 2 FUMIKAZU NAGASATO AND YOSHIKAZU YAMAGUCHI 2.2 or [CS83]). Here we recall some beautiful results using the SU(2)-character varieties shown by A. Casson and X.-S. Lin.Casson introduced an invariant for an integral homology 3-sphere Σ, so-called the Casson invariant, originally by using the intersection between SU(2)-character varieties associated to a Heegaard splitting of Σ. More precisely, the character varieties of two handlebodies associated to a Heegaard splitting of Σ are embedded in that of the common boundary surface of the handlebodies. Then the Casson invariant is defined as a half of the integer obtained by counting the algebraic intersection of the embedded character varieties of the handlebodies. (see [AM90, Sav99] for expositions). X.-S. Lin [Lin92] defined an invariant for a knot K in 3-sphere S 3 , now known as the Casson-Lin invariant, by applying similar idea to the SU(2)-character variety of the fundamental group of the knot exterior E K with trace free along meridians, whose representations send meridians to trace zero matrices. Lin showed that the Casson-Lin invariant is a half of the signature of the knot.Afterward, C. Herald generalized the Casson-Lin invariant by using gauge theoretic methods in [Her97] and M. Heusener and J. Kroll also have studied the same issue by topological methods in [HK98] to consider other trace condition of meridians indexed by t ∈ (−2, 2). The knot invariant by Herald's and Heusener-Kroll's generalization corresponds to the equivariant signature of the knot. On the other hand, O. Collin and N. Saveliev [CS01] have studied SU(2)-character varieties of knot exteriors with trace conditions of meridians from a viewpoint of gauge theory with cyclic group actions. They considered finite cyclic branched covers over integral homology 3-spheres with branch set a knot and define a topological invariant, called the equivariant Casson invariant, for integral homology 3-spheres with finite cyclic g...

This paper gives an explicit formula for the SL 2 (C)−non-abelian Reidemeister torsion as defined in [Dub06] in the case of twist knots. For hyperbolic twist knots, we also prove that the non-abelian Reidemeister torsion at the holonomy representation can be expressed as a rational function evaluated at the cusp shape of the knot. . , i.e. the word w is obtain from w by changing each of its letters by its reverse. Of course this choice is strictly equivalent to presentation (1). But in a sense, when m > 0 the word w m does not give a "reduced" relation (some cancelations are possible in w m xw −m y −1 ) which is not the case for the word Ω m . Some more elementary properties of twist knots are discussed in the following remark. 3 6 7

We give a surgery formula for the asymptotic behavior of the sequence given by the logarithm of the higher dimensional Reidemeister torsion. Applying the resulting formula to Seifert fibered spaces, we show that the growth of the sequences has the same order as the indices and we give the explicit values for the limits of the leading coefficients. There are finitely many possibilities as the limits of the leading coefficients for a Seifert fibered space. We also show that the maximum is given by −χ log 2 where χ is the Euler characteristic of the base orbifold for a Seifert fibered space. These limits of the leading coefficients give a locally constant function on a character variety. This function takes the maximum −χ log 2 only on the top-dimensional components of the SU(2)-character varieties for Seifert fibered homology spheres.