Natural properties of the trunk of a knot

TL;DR

This paper investigates the geometric complexity measure called the trunk of a knot, proving a key property about connected sums and exploring conjectures related to minimal embeddings.

Contribution

It proves that the trunk of a connected sum of knots equals the maximum of their individual trunks, confirming Ozawa's conjecture, and presents potential counterexamples to another conjecture.

Findings

Trunk of connected sum equals maximum of individual trunks

Confirmed Ozawa's conjecture on trunk behavior

Identified possible counterexamples to width-minimizing embeddings

Abstract

The trunk of a knot in $S^3$, defined by Makoto Ozawa, is a measure of geometric complexity similar to the bridge number or width of a knot. We prove that for any two knots $K_1$ and $K_2$, we have $tr(K_1 \# K_2) = \max\{tr(K_1),tr(K_2)\}$, confirming a conjecture of Ozawa. Another conjecture of Ozawa asserts that any width-minimizing embedding of a knot $K$ also minimizes the trunk of $K$. We produce several families of probable counterexamples to this conjecture.