Inscribed Trefoil Knots

TL;DR

This paper proves that for certain knots, one can inscribe a trefoil knot within a smooth analytic curve by connecting six specific points on the curve, linking polynomial invariants to geometric constructions.

Contribution

It establishes the existence of a six-point inscribed trefoil knot within analytic curves representing knots with nontrivial quadratic Conway polynomial terms.

Findings

Existence of six points forming a trefoil inscribed in analytic curves.

Connection between polynomial invariants and geometric inscribed knots.

Method applicable to knots with nontrivial quadratic Conway polynomial term.

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 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.