Inscribed Trefoil Knots

Abstract

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 that 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.