Chebyshev diagrams for rational knots

2009/06/22 by Pierre-Vincent Koseleff, Daniel Pecker, Koseleff, Pierre-Vincent +1
Mathematics · #11A55 #14H50 #14P99 #57M25 #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.0906.4083

openalex publication_date 2009/06/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that every rational knot K of crossing number N admits a polynomial parametrization x=T_a(t), y = T_b(t), z = C(t) where T_k(t) are the Chebyshev polynomials, a=3 and b+ °C = 3N. We show that every rational knot also admits a polynomial parametrization with a=4. If C (t)= T_c(t) is a Chebyshev polynomial, we call such a knot a harmonic knot. We give the classification of harmonic knots for a ≤ 4.

Related

• Chebyshev diagrams for two-bridge knots
• The first rational Chebyshev knots
• Computing Chebyshev knot diagrams
• Harmonic Knots
• Chebyshev Knots