### THE MD-ENERGY OF KNOTS AND "MING"

Ying-Qing Wu

This program and its related files are available via anonymous ftp from
ftp.math.uiowa.edu
in the directory of
wu/ming.

The MD energy is defined by J. Simon for polygonal
knots. A polygonal knot K consists of several edges E_1, ..., E_n in
the Euclidean space, which form a closed knotted loop. The ends of
the edges are called the vertices of the knot. The energy contributed
between E_i and E_j is

(L_i)(L_j)/(D_ij*D_ij)
where L_i is
the length of E_i, and D_ij is the minimum distance between E_i and
E_j. The energy of K is obtained by summing such contributions over
all E_i and E_j which are not adjacent.
A knot can be deformed to another by an "isotopy". The problem about
computation of knot energy is to find the minimal energy among all
knots isotopic to a given knot. "ming" is the a program which will try
to find the minimal energy by pushing the knot along the direction of
its "energy gradient". Actually this method does not find the minimal
energy. What it approaches are local minimals of the knot energy.
For example, the knots 4_1.8 and 4_1.8.2 are
both figure 8 knots with 8 edges, and their energy can not be reduced
by "ming", but their energies are very different: 228 with 304! No
algorithm is known to find the absolute minimum of the energy for a
given knot. Another suprising result of "ming" is that it founds a
trivial knot with 22 edges,
triv.min, which is apparently a local minimum of the unknot.

"ming" is the graphic version of the earlier program
"min". It can handle knots with up to 500 vertices. It uses
Silicon Graphics' Open Inventor to draw the knot pictures, so it can
run only on the SGI's (or at least so in our building.) Click here to
see some examples of the graphic
output of the program. One can specify different colors and sizes for
the knot.

Besides calculating MD energy of knots, MING can also be used to draw,
modify, and visualize knots. For more information about "ming", see
"commands of ming" and
"menu items of ming". Here is a postscript
manual for "ming".

Another program named
ked has been made by
Kenny Hunt. It can manipulate the knot, like adding or deleting
vertices, or moving a vertex around on the screen.

Please send any comments or suggestions to wu@math.uiowa.edu

to the Ying-Qing Wu's home page .