Knots are familiar, yet can be mysterious objects (especially when trying to untangle a messy one). Knowledge of some knots is a necessity for sailors; yet there is much that is not known about knots, and so mathematicians study “knot theory.”
Mathematical knots do not have two loose ends, like shoelaces that can be untied. To make the simplest mathematical knot, take a length of string (or flexible wire) and bend it so the two ends cross each other. Now take the end that is “on top” and twist it to go under, then over the other end. Finally, glue the two ends together. This is called a trefoil knot, and in its most symmetric presentation, looks like three identical rings woven together. It is impossible to undo this knot (or any mathematical knot) without cutting it.
When Anton gives instructions to his computer program to connect copies of a polyline generator to form closed circuits in space, some of the circuits among the thousands produced may be mathematical knots. The program contains a “filter” that can identify which of the circuits are knots, and Anton can select choose from these to have the basis of a knotted sculpture.