On the crossing number of high degree satellites of hyperbolic knots (Q1128278)

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 \(K\).