Smooth Knots with Odd Quadratic Term of the Conway Polynomial Have Inscribed Trefoils

Abstract

An inscribed knot is formed by polygonally connecting points lying on a knot $\gamma$ in parametric order, then closing the path by connecting the first and final points. The stick-knot number of a knot type K is the minimum number of line segments needed to polygonally form some knot with the same homotopy type. The stick-knot number of a trefoil is 6. Cole Hugelmeyer studied the manifold $M$ consisting of 6 points lying on a triangular prism and found that by intersecting a perturbation of $M'$, twisting the top of the prism, with $Q_\gamma$, the manifold of 6-tuples of points lying on $\gamma$, any analytic knot with nontrivial quadratic term of its Conway polynomial has an inscribed trefoil. We show that by using a perturbation of the double-cover of the orientation class $[M \cap Q_\gamma]$ and analysis of planar configurations, an analogous result holds for a class of smooth knots with odd quadratic term. We also show that in the analytic case, there are both inscribed left and right-handed trefoils.