Chebyshev diagrams for two-bridge knots

Details Chebyshev diagrams for two-bridge knots Chebyshev diagrams for two-bridge knots

2009-09-17

arxiv.org/abs/0909.3281v1

Mathematics

Geometric Topology

25p. 17 figures, submitted. This is a part of "Chebyshev diagrams for rational knots", arXiv:0906.4083

Scientific paper

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.

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