These are some results and figures by
University of Iowa
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?
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.
c[K] is bounded by 2*(d[K])^2
d[K] is bounded by 2*(c[K]+1)^2
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)
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)
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