Natural properties of the trunk of a knot

Abstract

The trunk of a knot in [Formula: see text], 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 [Formula: see text] and [Formula: see text], we have [Formula: see text], confirming a conjecture of Ozawa. Another conjecture of Ozawa asserts that any width-minimizing embedding of a knot [Formula: see text] also minimizes the trunk of [Formula: see text]. We produce several families of probable counterexamples to this conjecture.