Trivial or Knot: A Software Tool and Algorithms for Knot Simplification

A special type of representation for knots and for local knot manipulations is described and used in a software tool called TOK to implement a number of algorithms on knots. Two algorithms for knot simplification are described: simulated annealing applied to the knot representation, and a “divide-simplify-join” algorithm. Both of these algorithms make use of the compact knot representation and of the basic mechanism TOK provides for carrying out a predefined knot manipulation on the knot representation. The simplification algorithms implemented with the TOK system exploit local knot manipulations and have proven themselves effective for simplifying even very complicated knots in reasonable time. Introduction What is Knot Theory? Knots are very complicated mathematical objects that have intuitive, real-world counterparts. This makes them very interesting to study. A tangle in a (frictionless) rope is a knot if when the ends of the rope are pulled in opposite directions, the tangle is not unraveled. Given a pile of rope with two ends sticking out, it is difficult, or even impossible to say by inspection whether or not the rope is truly knotted. An even more difficult problem is to decide if two piles of tangled rope are equivalent; meaning that one pile may be stretched and deformed to look like the other pile without tearing the rope. Figure 1 illustrates that equivalence is sometimes not obvious even for simple knots. Figure 1. (a) Two trefoil knots (b) Two trivial knots Knot theory studies an abstraction of the intuitive “knot on a rope” notion. The theory deals with questions such as proving knottedness, and classifying types of knottedness. In a more abstract sense we may say that knot theory studies the placement problem: “Given spaces X and Y, classify how X may be placed in Y”. Here a placement is usually an embedding, and classification often means up to some form of movement. In these terms classical knot theory studies embeddings of a circle in Euclidean three space. (Hence we consider the two ends of the rope tied together) There are two main schools in knot theory research. The first is called combinatorial or pictorial knot theory. Here the main idea is to associate with the mathematical object a drawing that represents the knot, and to study various combinatorical properties of this drawing. The second school considers the abstract notion of a knot as an embedding and studies the topology of the so called complementary space of the image of the embedding, by applying to this space the tools of Algebraic Topology. This paper dwells in the first realm pictorial knot theory. Following is a brief description of the basic theory that is needed to understand the TOK knot manipulation tool. For a more comprehensive overview see [1][2][3]. * Electrical Engineering Department Technion, 3200 Haifa, Israel [email protected] [email protected] ** Computer Science Department Technion, 3200 Haifa, Israel (on sabbatical at AT&T Bell Laboratories, Murray Hill, NJ 07974, USA)