On the crossing number of high degree satellites of hyperbolic knots
It is believed that the crossing number \(\text{cr} (K')\) of a satellite knot \(K'\) is at least equal to that of a companion, and that the crossing number of a degree \(p\) satellite \(K'\) of \(K\) is at least \(p^2\) times the crossing number of \(K\). In previous work Freedman and the author showed that \(\text{cr} (K')\geq p^2(2 \text{genus} (K)-1)\). In the present paper, satellites of hyperbolic knots are considered. It is shown that \(\text{cr} (K')\) is bigger or equal to \(p^2\) times a constant depending on the area and the length of the meridian \(m\) of the ``critical horo-torus'' \({\mathcal E} (K)\) of the complement of the hyperbolic knot \(K\). This is an immersed Euclidean torus which, by definition, is covered by the smallest horo-sphere in the universal covering whose projection is not an embedded torus. The exact formula is \(\text{cr} (K')\geq p^2 (\text{area} {\mathcal E} (K)/(\text{length} ([m]) (2\pi- \text{length} ([m]))))\). Using Weeks' computer program SnapPea this is applied to various Montesinos knots where the estimate turns out to improve the above estimate involving the genus of the knot \(K\).
scientific article; zbMATH DE number 1187597