The first rational Chebyshev knots

A Chebyshev knot C(a, b, c, ) is a knot which has a parametrization of the form x(t) = Ta(t); y(t) = Tb(t); z(t) = Tc(t + ), where a, b, c are integers, Tn(t) is the Chebyshev polynomial of degree n and 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 C(a, b, c, ). We deduce a list of minimal Chebyshev representations of two-bridge knots with small crossing number.

• Type: Journal
• Year: 2010
• Where: JSC
• Authors: Pierre-Vincent Koseleff, D. Pecker, F. Rouillier
• Keywords: Chebyshev Knots, JSC 2010, Minimal Chebyshev Representations, Two-bridge Knots