A computer program called SnapPea, written by Jeffrey Weeks of the NSF-sponsored Geometry Center located at the University of Minnesota, can be used to analyze so-called geometric structures on certain three-dimensional spaces, such as the complement of a knot in ordinary three-space. One can start by drawing a picture of a knot, such as the "figure eight knot" (left), using a mouse. The program then investigates the existence of a geometric structure on the complement of the knot, and computes a number of different numerical invariants of the space, such as its volume. It also produces diagrams that describe pictorially the behavior of the space as one approaches "infinity." The diagram at right--in technical language, the horoball diagram--is associated with the complement of the figure eight knot. Part of the research of Allan Edmonds, Professor of Mathematics, Indiana University Bloomington, has included the study of symmetries of knots. The associated horoball diagram has proved to be an important tool in that study, as one can see that symmetries of the knot are displayed as symmetries in the more rigid horoball diagram.

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