Chebyshev diagrams for two-bridge knots

TL;DR

This paper demonstrates that all two-bridge knots with crossing number N can be parametrized using Chebyshev polynomials, introducing harmonic knots and classifying them for specific cases.

Contribution

It introduces a polynomial parametrization for two-bridge knots using Chebyshev polynomials and classifies harmonic knots for certain parameters.

Findings

Every two-bridge knot of crossing number N admits a Chebyshev polynomial parametrization.

Harmonic knots are classified for the case when C(t) is a Chebyshev polynomial with a ≤ 3.

Most results are derived from continued fractions and matrix representations.

Abstract

We show that every two-bridge knot $K$ of crossing number $N$ admits a polynomial parametrization $x=T_3(t), y = T_b(t), z =C(t)$ where $T_k(t)$ are the Chebyshev polynomials and $b+\deg C = 3N$. If $C (t)= T_c(t)$ is a Chebyshev polynomial, we call such a knot a harmonic knot. We give the classification of harmonic knots for $a \le 3.$ Most results are derived from continued fractions and their matrix representations.