Inscribed Trefoil Knots

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 $0\le t_1<t_2<\mathrm{.}\mathrm{.}\mathrm{.}<t_6<1$ 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.