Character varieties and peripheral polynomials of a class of knots

Abstract

The representation space or character variety of a finitely generated group is easy to define but difficult to do explicit computations with. The fundamental group of a knot can have two interesting representations into PSL2(C) coming from oppositely oriented complete hyperbolic structures. These two representations lift to give four excellent SL2(C) representations. The excellent curves of a knot are the components of the SL2(C) character variety containing the excellent representations. It is possible to compute geometric invariants of hyperbolic cone manifolds from suitable descriptions of the excellent curve. In this paper, Hilden, Lozano and Montesinos describe a method for analyzing the character varieties of a large class of knots. The main ingredients in this method are a non-obvious, but convenient parametrization of 2×2 complex matrices and an explicit computation relating the holonomies of the four punctures of a four punctured sphere. In order to qualify when their method will work, Hilden, Lozano and Montesinos introduce the notion of a 2n-net. A 2n-net is an interesting generalization of a 2n-plat. Recall that a 2n-plat is obtained by separately closing the top and the bottom of a 2n-strand braid. A 2n-net is the generalization obtained by allowing rational tangles at the crossing points. The given method to analyze the character variety works for any knot with a 4-net description. The method is remarkably robust. For example, it works for essentially every knot in the table in D. Rolfsen's book [Knots and links, Publish or Perish, Berkeley, Calif., 1976