On the character variety of periodic knots and links

For a hyperbolic knot, the excellent component of the character curve is the one containing the complete hyperbolic structure on the complement of the knot. In this paper we explain a method to compute the excellent component of the character variety of periodic knots. We apply the method to those knots obtained as the preimage of one component of a 2-bridge link by a cyclic covering of S-3 branched on the other component. We call these knots periodic knots with rational quotient. Among this class of knots are the 'Turk's head knots'. Finally we give some invariants deduced from the excellent component of the character curve, such as the h-polynomial and the limit of hyperbolicity for all the periodic knots with rational quotient, up to 10 crossings, which are not 2-bridge or toroidal.