The first rational Chebyshev knots

2009/11/03 by Koseleff, Pierre-Vincent, Pecker, Daniel, Rouillier, Fabrice
#14H50 #14P99 #57M25 #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.0911.0566

Abstract

A Chebyshev knot \cal C(a,b,c,ϕ) is a knot which has a parametrization of the form x(t)=T_a(t); y(t)=T_b(t) ; z(t)= T_c(t + ϕ), where a,b,c are integers, T_n(t) is the Chebyshev polynomial of degree n and ϕ∈ \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 \cC(a,b,c,ϕ). We deduce a list of minimal Chebyshev representations of two-bridge knots with small crossing number.

Related

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