Chebyshev diagrams for two-bridge knots

Pierre-Vincent Koseleff (1, 2) , Daniel Pecker (1)

1 UPMC - Université Pierre et Marie Curie - Paris 6
2 SALSA - Solvers for Algebraic Systems and Applications

We show that every two-bridge knot $K$ of crossing number $N$ admits a polynomial parametrization $x=T_3(t), \, y = T_b(t), z =C(t)$ where $T_k(t)$ are the Chebyshev polynomials and $b+\deg C = 3N$. 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 $a \le 3.$ Most results are derived from continued fractions and their matrix representations.

Mots clés: continued fractions.; rational knots; continued fractions; Polynomial curves; Chebyshev polynomials; Chebyshev curves

Domaines: Topologie géométrique [math.GT]

Pierre-Vincent Koseleff, Daniel Pecker. Chebyshev diagrams for two-bridge knots. Geometriae Dedicata, 2011, 150 (1), pp.405-425. ⟨10.1007/s10711-010-9514-7⟩. ⟨hal-00418298⟩