# Trunk of Satellite and Companion Knots

## TL;DR

This paper investigates the relationship between the trunk invariant of satellite knots and their companion knots, establishing bounds involving winding and wrapping numbers, and extends these results to more general cases.

## Contribution

It provides new inequalities relating the trunk of satellite knots to that of their companions, incorporating winding and wrapping numbers, and discusses potential generalizations.

## Key findings

- Established that trunk(K) ≥ n · trunk(J) for satellite knot K with companion J and winding number n.
- Showed that trunk(K) > (1/2) m · trunk(J) when considering wrapping number m.
- Discussed possible extensions of the main inequalities to broader contexts.

## 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 ${\rm trunk}(K) \geq n \cdot {\rm trunk}(J)$ where ${\rm trunk}(\cdot)$ 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 ${\rm trunk}(K) > \frac{1}{2} m\cdot {\rm trunk}(J)$ since $m \geq n$. We also discuss generalizations of the second result.