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\leq 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.