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We show that every nonzero integer occurs as 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 \(n \ge 4m + 1\), 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.

Denominators and differences of boundary slopes for (1,1)-knots