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...

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We show that all twist knots, certain double twist knots and some other 2-bridge knots are minimal elements for the partial ordering on the set of prime knots.The key to these results are presentations of their character varieties using Chebyshev polynomials and a criterion for irreducibility of a polynomial of two variables. These give us an elementary method to discuss the number of irreducible components of the character varieties, which concludes the result essentially.

In this paper, we first give a diagrammatic analogue of the Young symmetrizer. By using this, the ðslðN; CÞ; Þ-weight system for an arbitrary finite-dimensional irreducible representation is formulated in a diagrammatic way. The formula is useful for the calculations of the ðslðN; CÞ; Þ-weight system in the sense that we do not need actual constructions of the representations of slðN; CÞ essentially. Hence by using this and the modified Kontsevich integral we can get the quantum ðslðN; CÞ; Þ-invariant for any finite-dimensional irreducible representation without actual constructions of the representations of slðN; CÞ. The diagrammatic construction is a generalization of the formula given in ''Remarks on the ðslðN; CÞ; adÞ-weight system''.

In this paper we give a formula for the A-polynomial of the (2, 2p + 1)-torus knot, for any integer p, by using noncommutative methods (the Kauffman bracket skein modules).