Natural properties of the trunk of a knot

2016/07/29 by Davies, Derek, Zupan, Alexander
#57M25 #57M27 #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1608.00019

Abstract

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) = max\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.

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