Chebyshev diagrams for two-bridge knots

2009/09/17 by Koseleff, Pierre-Vincent, Pecker, Daniel
#14H50 #14P99 #57M25 #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.0909.3281

Abstract

We show that every two-bridge knot K of crossing number N admits a polynomial parametrization x=T₃(t), y = T_b(t), z =C(t) where T_k(t) are the Chebyshev polynomials and b+°C = 3N. 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 ≤ 3. Most results are derived from continued fractions and their matrix representations.

Related

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