Trunk of Satellite and Companion Knots

Abstract

We study the knot invariant called trunk, as defined by Ozawa, and the relation of the trunk of a satellite knot with the trunk of its companion knot. Our first result is $\mathrm{trunk}(K)\ge n\cdot \mathrm{trunk}(J)$ where $\mathrm{trunk}(\cdot )$ denotes the trunk of a knot, $K$ is a satellite knot with companion $J$, and $n$ is the winding number of $K$. To upgrade winding number to wrapping number, which we denote by $m$, we must include an extra factor of $\frac{1}{2}$ in our second result $\mathrm{trunk}(K)>\frac{1}{2}m\cdot \mathrm{trunk}(J)$ since $m\ge n$. We also discuss generalizations of the second result.

21 pages, 5 figures