MING -- WHAT IS IT

Ying-Qing Wu

The MING program has the following functions:

  • Minimize MD energy of polygonal knots;
  • Reduce number of edges representing a polygonal knot;
  • Plot and manipulate polygonal knots.

    • Knot energy minimizer.

    The main purpose of "MING" is to find local minimals for the MD (minimum distance) energy function of polygonal knots. The MD energy of a polygonal knot is defined by J. Simon in our department.

    A polygonal knot K consists of several edges E1, ..., En 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 Ei and Ej is

    Li Lj / Dij2
    where Li is the length of Ei, and Dij is the minimum distance between Ei and Ej. The energy of K is obtained by summing such contributions over all Ei and Ej 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 two figure 8 knots in the picture both have 8 edges. Their energy can not be reduced by "ming" and are apparently local minimals, 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 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.) One can specify different colors and sizes for the knot.

    Edge number reducer

    The minimum edge number of a knot is the minimum number of edges one can use to build the knot. Professor Richard Randell in our department is an expert in this area. One method to find the minimum edge number of a knot is to represent the knot as a polygonal knot, then reduce the number of edges by deleting unnecessary edges.

    "Ming" can be used to attempt reducing the number of edges for a polygonal knot. The idea is as follows. Given a polygonal knot, it looks at all triangles spanned by two adjacent edges to see if any of these is disjoint from other edges of the knot. If that is the case, then these two edges can be replaced by a single edge without changing the knot type. If that fails, the program puts an extra "external force" on the ends of one of the edges, then use the energy minimizer to modify the knot to an optimal position. The extra force added has the effect of trying to shrink that edge to a point. This would often produces a new configuration in which the squeezed edge can be removed.

    Knot drawer and manipulator

    Besides calculating MD energy of knots, MING can also be used to draw, modify, and visualize knots. See the section on Drawing and visualizing knots for more details.

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