Boundary slopes and the numbers of positive/negative crossings for Montesinos knots (Q2900335)

In the exterior of a knot, a properly embedded essential surface provides a set of parallel simple closed curves on the boundary torus, which can be identified by an irreducible fraction, possibly \(1/0\), called the numerical boundary slope of the surface.NEWLINENEWLINEA Montesinos link is obtained by connecting rational tangles, defined by rational numbers \(P_i/Q_i\), \(i=1, \dots , N\). Assume that \(N\geq 3\) and each \(P_i/Q_i\) is a non-integral non-infinity fraction. Then the main result of this paper (Theorem~1.1) is the following: let \(D\) be a standard diagram of a Montesinos knot \(K\) and assume that \(D\) has \(c_+\) positive crossings and \(c_-\) negative crossings. Then \(-2c_-\leq R \leq 2c_+\) for any finite numerical boundary slope \(R\) of \(K\). Here standard diagram (of a Montesinos knot) means a specific diagram constructed in a natural way. Since the standard diagram can be chosen attaining the minimal number of crossings \(c(K)\) of the knot, it follows that Diam\((K)\leq 2c(K)\) where Diam\((K)\) is the maximum minus the minimum among all the finite numerical boundary slopes.NEWLINENEWLINEThe proof is based on a careful examination of the algorithm by \textit{A. E. Hatcher} and \textit{U. Oertel} [Topology 28, No. 4, 453--480 (1989; Zbl 0686.57006)] which completely determines the set of boundary slopes for a Montesinos knot.