Boundary slopes for Montesinos knots

Abstract

FOR A KNOT K c S3, let S(K) c Q u {CQ} be the set of slopes of boundary curves of incompressible, %incompressible orientable surfaces in the knot exterior, slopes being normalized in the standard way so that a longitude has slope 0, a meridian slope co. These sets S(K) of %slopes are of special interest because of their relation with Dehn surgery and character varieties; see e.g., [2]. The only general results known so far are that S(K) is always finite [S] and contains at least two elements [3], including of course 0 (coming from a minimal genus Seifert surface). Only for special classes of knots has S(K) been determined exactly. For the (p, q) torus knot, S(K) = (0, pq}. For 2-bridge knots, S(K) is an arbitrarily large set of even integers computable via continued fractions [7], A non-integer s for the motivating example (- 2,3,7) we find S(K) = (0, 16, 18; .20). For somewhat more complicated cases, a small computer can do the work rather quickly. Some examples of these computer calculations are given in the last section of the paper. These include the Montesinos knots of I 10 crossings, plus a few other random examples of greater complexity.