Computing Chebyshev knot diagrams

TL;DR

This paper develops an efficient method to compute all possible knot diagrams of Chebyshev polynomial knots by analyzing how the diagram varies with the phase parameter, using a complexity of roughly quadratic in the product of the degrees.

Contribution

It provides a systematic approach to list all knot diagrams of Chebyshev polynomial knots with specific conditions, improving computational efficiency.

Findings

All possible knot diagrams can be listed in (n^2) bit operations.

The method applies when a is odd and coprime with b.

The approach characterizes knot diagram variations as (n^2) complexity.

Abstract

A Chebyshev curve $\mathcal{C}(a,b,c,\phi)$ 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 polynomialof degree $n$ and $\phi \in \mathbb{R}$. When $\mathcal{C}(a,b,c,\phi)$ is nonsingular,it defines a polynomial knot. We determine all possible knot diagrams when $\phi$ varies. Let $a,b,c$ be integers, $a$ is odd, $(a,b)=1$, we show that one can list all possible knots $\mathcal{C}(a,b,c,\phi)$ in$\tilde{\mathcal{O}}(n^2)$ bit operations, with $n=abc$.