The first rational Chebyshev knots

TL;DR

This paper demonstrates that all two-bridge knots can be represented as Chebyshev knots with specific parameters and provides algorithms to find such representations, including minimal forms for knots with small crossing numbers.

Contribution

It establishes that any two-bridge knot can be parametrized as a Chebyshev knot with a=3 or 4 and introduces algorithms to find all such representations for given parameters.

Findings

All two-bridge knots are Chebyshev knots with a=3 or 4.

Algorithms are provided to compute Chebyshev representations for given parameters.

Minimal Chebyshev forms for small crossing number knots are listed.

Abstract

A Chebyshev knot ${\cal C}(a,b,c,\phi)$ is a knot which has a parametrization of the form $ x(t)=T_a(t); y(t)=T_b(t) ; z(t)= T_c(t + \phi), $ where $a,b,c$ are integers, $T_n(t)$ is the Chebyshev polynomial of degree $n$ and $\phi \in \R.$ We show that any two-bridge knot is a Chebyshev knot with $a=3$ and also with $a=4$. For every $a,b,c$ integers ($a=3, 4$ and $a$, $b$ coprime), we describe an algorithm that gives all Chebyshev knots $\cC(a,b,c,\phi)$. We deduce a list of minimal Chebyshev representations of two-bridge knots with small crossing number.