Computing Chebyshev knot diagrams

Abstract

A Chebyshev curve C(a,b,c,φ) has a parametrization of the form x(t)=Ta(t); y(t)=Tb(t); z(t)=Tc(t+φ), where a,b,c are integers, Tn(t) is the Chebyshev polynomial of degree n and φ∈R. When C(a,b,c,φ) is nonsingular, it defines a polynomial knot. We determine all possible knot diagrams when φ varies. When a,b,c are integers, (a,b)=1, we show that one can list all possible knots C(a,b,c,φ) in O˜(n2) bit operations, with n=abc. We give the parameterizations of minimal degree for all two-bridge knots with 10 crossings and fewer.