Matrix Models an Knot Theory

We shall explain how knot, link and tangle enumeration problems can be expressed as matrix integrals which will allow us to use quantum field-theoretic methods. We shall discuss the asymptotic behaviors for a great number of intersections. We shall detail algorithms used to test our conjectures. 1. Classification and Enumeration of Knots, Links, Tangles A knot is defined as a closed, non-self-intersecting curve that is embedded in three dimensions and cannot be untangled to produce a simple loop (i.e., the unknot). A knot can be represented by its plane projection (i.e., its diagram). A knot can be generalized to a link, which is simply a knotted collection of one or more closed strands. A tangle is defined as a region in a knot or link projection plane surrounded by a circle such that the knot or link crosses the circle exactly four times. An alternating knot (resp. link) is a knot (resp. link) which possesses a knot diagram (resp. link diagram) in which crossings alternate between underand overpasses (see Figure 1). Figure 1. An alternating link, a tangle and an non-alternating knot Figure 2. A 61 knot of the Tait’s classification P. G. Tait [16, 17, 18, 19, 20, 21] undertook a study of knots in response to Kelvin’s conjecture that the atoms were composed of knotted vortex tubes of ether (Thompson [24]). He categorized knots in terms of the number of crossings in a plane projection (see Figure 2). He also made some conjectures which remained unproven until the discovery of Jones polynomials: 14 Matrix Models an Knot Theory 1. Reduced alternating diagrams have minimal link crossing number, 2. Any two reduced alternating diagrams of a given knot have equal writhe, 3. The flyping conjecture, which states that the number of crossings is the same for any reduced diagram of an alternating knot (see [25] for definition of the flyping equivalence). Conjectures (1) and (2) were proved by Kauffman [4], Murasugi [9], and Thistlethwaite [22, 23] using properties of the Jones polynomial or Kauffman polynomial F (see Hoste et al. [1]). Conjecture (3) was proved true by Menasco and Thistlethwaite [7, 8] using properties of the Jones polynomial. Schubert [12] showed that every knot can be uniquely decomposed (up to the order in which the decomposition is performed) as a knot sum of a class of knots known as prime knots, which cannot themselves be further decomposed. Knots that are the sums of prime knots are known as composite knots. There is no known formula for giving the number of distinct prime knots as a function of the number of crossings. The numbers of distinct prime knots having n = 1, 2, . . . crossings are 0, 0, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, . . . (Sloane’s M0851) [13]. In the 1932, Reidemeister [10] first rigorously proved that knots exist which are distinct from the unknot. He did this by showing that all knot deformations can be reduced to a sequence of three types of “moves,” called Reidemeister moves (see Figure 3).