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Denominators and differences of boundary slopes for \((1,1)\)-knots. (English) Zbl 1353.57006

Summary: 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 \(n \geq 4m+1\), 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.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
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