Properties of knots preserved by cabling

We examine geometric properties of a knot $J$ that are unchanged by taking a $(p,q)$-cable $K$ of $J$. Specifically,we show that $w(K) = q^2 \cdot w(J)$, where $w(K)$ is the width of $K$ in the sense of Gabai. We use this fact todemonstrate that thin position is a minimal bridge position of $J$ if and only if the same is true for $K$, and moregenerally we show that an “obvious” cabling of any thin position of $J$ is a thin position of $K$. We conclude byproving that $J$ is meridionally small (mp-small) if and only if $K$ is meridionally small (mp-small), and if $J$ ismp-small and every non-minimal bridge position for $J$ is stabilized, then the same is true for~$K$.

Full Text (PDF format)

Published 3 October 2011