Harmonic Knots

Details Harmonic Knots Harmonic Knots

2012-03-20

arxiv.org/abs/1203.4376v1

Mathematics

Geometric Topology

24 p., 47 fig

Scientific paper

The harmonic knot $H(a,b,c)$ is parameterized as $K(t)= (T_a(t), T_b (t), T_c (t))$ where $a$, $b$ and $c$ are relatively coprime integers and $T_n$ is the degree $n$ Chebyshev polynomial of the first kind. We classify the harmonic knots $H(a,b,c)$ for $ a \le 4$. We show that the knot $H(2n-1, 2n, 2n+1)$ is isotopic to $H(4,2n-1, 2n+1)$ (up to mirror symmetry). We study the knots $H(5,n,n+1)$ and give a table of the simplest harmonic knots.

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