Computing Chebyshev knot diagrams

TL;DR

This paper characterizes all possible knots formed by Chebyshev curves with given degrees, providing a comprehensive classification of knot types based on their polynomial parametrizations.

Contribution

It explicitly determines all knots realizable by Chebyshev curves with fixed degrees, advancing the understanding of polynomial knot representations.

Findings

Classified all knots from Chebyshev curves with given degrees

Identified conditions for the absence of double points

Provided a complete enumeration of such polynomial knots

Abstract

A Chebyshev curve C(a,b,c,\phi) has a parametrization of the form x(t)=Ta(t); y(t)=T_b(t) ; z(t)= Tc(t + \phi), where a,b,c are integers, Tn(t) is the Chebyshev polynomial of degree n and \phi \in \RR. When C(a,b,c,\phi) has no double points, it defines a polynomial knot. We determine all possible knots when a, b and c are given.