  We show that every nonzero integer occurs in the denominator of a boundary
slope for infinitely many (1,1)-knots and that infinitely many (1,1)-knots have
boundary slopes of arbitrarily small difference. Specifically, we prove that
for any integers m, n > 1 with n odd the exterior of the Montesinos knot
K(-1/2, m/(2m \pm 1), 1/n) in S^3 contains an essential surface with boundary
slope r = 2(n-1)^2/n if m is even and 2(n+1)^2/n if m is odd. If n > 4m, we
prove that K(-1/2, m/(2m+1), 1/n) also has a boundary slope whose difference
with r is (8m-2)/(n^2-4mn+n), which decreases to 0 as n increases. All of these
knots are (1,1)-knots.
