Boundary Slopes for Montesinos Knots

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 &slope was found first for the (-2, 3, 7) pretzel knot, where by combining results of CullerGordon-Luecke-Shalen, Fintushel-Stern, and Oertel it was observed that there must be a fractional &lope between 18 and 19. Other instances of non-integer &lopes have been found subsequently by Takahashi [ 131. In this paper we describe an algorithm for computing S(K) for a class of knots which is the “vector sum” of 2-bridge and pretzel knots, the Montesinos knots K(p,/q,, . . . , p./q.) obtained by connecting n rational tangles of slopes pl/ql, . . . , pn/qn in a simple cyclic pattern. In this notation 2-bridge knots are the cases n < 2, while the (ql, . . . , 4.) pretzel knot is K(l/q,, . . . , l/q,). The algorithm is quite effective. Simple cases like the (-2,3,7) pretzel, or more generally any (ql, q2, q3) pretzel, can easily be done by hand; 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. Non-integer %slopes occur quite often, starting with the first non-torus, non-alternating knot in the tables, 8,,, where 8/3 E S(K). For more complicated Montesinos knots, %slopes occur in such abundance that it seems difficult to find general patterns in the sets S(K). We produce examples showing that all rational numbers occur among the S-slopes of Montesinos knots. On the other hand, we show that Montesinos knots having an easily seen alternating projection, namely the ones with all pi/qi’s of the same sign, have all &slopes even integers. Perhaps this is a general property of alternating knots. To compute S(K) we in fact determine fairly explicitly all the incompressible surfaces with non-empty, non-meridional boundary in the exterior of K = K(p,/q,, . . . ,pJq.). (For meridional boundary it was shown in [ 1 I] that cc E S(K) if and only if n 2 4, assuming without loss of generality that /qil > 2 for each i.) Incompressible surfaces in rational tangles having been analyzed thoroughly in [7], our main task is understanding how to fit together incompressible surfaces in the separate tangles so as to form a surface in S3 K which is still