On the character variety of tunnel number 1 knots

Abstract

Given a hyperbolic knot K in S3, the SL2(C) characters ofπ1(S3−K) form an algebraic variety Cˆ(K). The algebraic component containing the character of the complete hyperbolic structure of S3−K is an algebraic curve CˆE(K). The desingularization of the projective curve corresponding to CˆE(K) is a Riemann surface Σ(K), and the trace function corresponding to the meridian of K induces a map p:Σ(K)→C. The pair (Σ(K),p) contains a great deal of information about the knot K and its hyperbolic structure. It can be described by a polynomial rE[K](y,z). There is an algebraic number yh which is a particular critical point of p in the interval (−2,2). It defines an angle 0<αh<2π with yh=2cos(αh/2), called the limit of hyperbolicity. The minimal polynomial hK(y) of yh is called the h-polynomial of K. The calculation of these invariants is in general quite complicated. In this paper the authors develop a method to calculate rE[K](y,z) and hK(y) for any tunnel number one knot, and they apply the method to the knots 10139 and 10161.