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

TL;DR

This paper proves that certain smooth knots with specific algebraic properties have inscribed trefoils, extending previous results from analytic knots using topological and geometric perturbation methods.

Contribution

It generalizes the existence of inscribed trefoils to a class of smooth knots with odd quadratic terms in their Conway polynomial, building on prior work with analytic knots.

Findings

Smooth knots with odd quadratic Conway polynomial term have inscribed trefoils.

Both left and right-handed inscribed trefoils exist in the analytic case.

Extension of previous topological methods to smooth knots with similar properties.

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.