Inscribed Trefoil Knots

Abstract

Let $K$ be a knot type for which the quadratic term of the Conway polynomial is nontrivial, and let $\gamma :\mathbb{R}\to {\mathbb{R}}^3$ be an analytic $\mathbb{Z}$-periodic function with non-vanishing derivative which parameterizes a knot of type $K$ in space. We prove that there exists a sequence of numbers $0\le t_1<t_2<...<t_6<1$ so that the polygonal path obtained by cyclically connecting the points $\gamma (t_1),\gamma (t_2),...,\gamma (t_6)$ by line segments is a trefoil knot.