Natural properties of the trunk of a knot

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The trunk of a knot in S³, 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 K₁ and K₂, we have tr(K₁ # K₂) = {tr(K₁),tr(K₂)}, 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.