These are some results and figures by
Aaron Trautwein
Ph.D. Thesis
HARMONIC KNOTS
University of Iowa
May, 1994

Each knot type can be represented by a parametrized curve in 3-space [x(t), y(t), z(t) ], where each of x, y, z is a finite trigonometric polynomial (i.e. a truncated Fourier series).
The degree of such a parametrization is the largest degree of any term appearing in any of the three equations.

The harmonic degree of a knot type [K] is the smallest degree parametrization of any knot in that knot type, denoted d[K] .

The basic question is: Which knots have which degrees?


Theorem: d[K] is greater than or equal to the super-bridge index of [K], which in turn is (Kuiper) strictly greater than the bridge index of [K].

Corollary: If K is a nontrivial knot, then d[K] > 2.
In particular, a nontrivial knot requres degree three or more for a trigonometric parametrization.
Corollary A composite knot required degree 4 or more.

Theorem: The harmonic degree and minimum crossing numbers of [K] give bounds on each other. That is, a parametrization of given degree cannot have too many crossings, and, on the other hand, given a knot with some number of crossings, one can bound the degree needed to represent that knot type.

Specifically,
c[K] is bounded by 2*(d[K])^2
and
d[K] is bounded by 2*(c[K]+1)^2


Here are explicit, minimum degree, parametrizations for the trefoil and figure-eight knots.
Nicer coefficients for these two, and all other knots through 8 crossings are available from Trautwein.

trefoil

x(t) = .41cos(t)-. 18sin(t)-.83cos(2t)-.83sin(2t)-.11cos(3t)+.27sin(3t)
y(t) = .36cos(t)+.27sin(t)- 1.13cos(2t)+.30sin(2t)+.11 cos(3t)-.27sin(3t)
z(t) = .45sin(t)-.30cos(2t)+1.13sin(2t)-.11cos(3t)+.27sin(3t)

figure-eight

x(t) = .32cos(t)-.51sin(t)-1.04cos(2t)-.34snn(2t)+1.04cos(3t)-.91sin(3t)
y(t) = .94cos(t).+41sin(t)+1.13cos(2t)-.68cos(3t)-1.24sin(3t)
z(t) = .16cos(t)+.73sin(t)-2.11cos(2t)-.39sin(2t)-.99cos(3t)-.21sin(3t)


Here are pictures of "harmonic knots", knots parametrized as trigonometric polynomials of low degree.
The trefoil, figure-eight, five(sub)one, and granny are known to be minimum degree; the others are not certain.


For more information, see to Aaron Trautwein's home page
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