Bounds on numerical boundary slopes for Montesinos knots

Abstract

We give an upper bound on the denominators of numerical boundary slopes and an upper bound on the differences between two numerical boundary slopes for Montesinos knots.

Pairs of boundary slopes with small differences

“…In general, plural candidate surfaces correspond to an edgepath system, but the twist is well defined. See [6] or [7]. Under these settings, as shown in [6], or as explained in [7], we can calculate the boundary slope of a candidate surface as follows.…”

“…See [6] for a fundamental reference, and also see [7][8][9][10][11] and [1] for related results and detailed explanations.…”

Crosscap numbers of pretzel knots

“…By boundary compressions, as argued in the proof of Theorem 1 in [3], an essential surface F E is obtained. Then it satisfies −χ (F E ) < −χ (F C ), and together with χ (F E ) 0 obtained in [4] and s(F C ) = 1…”

“…In the following, we will assume that the readers are rather familiar with the work of Hatcher and Oertel. See [2] or [4] for details.…”

“…As far as the authors know, results on the determination of crosscap numbers are only as follows: A knot has crosscap number one if and only if it is (2, n)-cabled, shown by Clark [1]. The knot 7 4 in the knot table has crosscap number three; Crosscap(7 4 ) = 3, proved by Murakami and Yasuhara [5]. This gives the first example showing that the inequality Crosscap(K ) 2g(K ) + 1, presented by Clark, is sharp.…”