### 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.