Mathematical Research Letters

Volume 5 (1998)

Number 2

On the crossing number of High Degree Satellites of Hyperbolic Knots

Pages: 235 – 246



Zheng-Xu He (U.C. San Diego)


Let $K$ be a hyperbolic knot, and let $K'$ be a satellite of $K$ of (homological) degree $p$, where $p$ is an integer. We show that the crossing number of $K'$ is at least $\big({\area(\E)}\big)\big({\len([m])\big)^{-1}\big(2\pi-2\len([m])}\big)^{-1} p^2$, where $\area(\E)$ is the area of the critical horo-torus of the hyperbolic structure on the knot complement and $\len([m])$ is the length of the meridian in the horo-torus. Our estimate is an improvement over an earlier result of M. Freedman and the author in many cases.

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