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? **

**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

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)

The trefoil, figure-eight, five(sub)one, and granny are known to be minimum degree; the others are not certain.

to
Aaron Trautwein's home page

to
Jonathan Simon's home page