  Let X be a norm curve in the SL(2,C)-character variety of a knot exterior M.
Let t = || b || / || a || be the ratio of the Culler-Shalen norms of two
distinct non-zero classes a, b in H_1(\partial M, Z). We demonstrate that
either X has exactly two associated strict boundary slopes \pm t, or else there
are strict boundary slopes r_1 and r_2 with |r_1| > t and |r_2| < t. As a
consequence, we show that there are strict boundary slopes near cyclic, finite,
and Seifert slopes. We also prove that the diameter of the set of strict
boundary slopes can be bounded below using the Culler-Shalen norm of those
slopes.
