Invariants of Random Knots and Links

Abstract

We study random knots and links in R 3 using the Petaluma model, which is based on the petal projections developed in [2]. In this model we obtain a formula for the limiting distribution of the linking number of a random two-component link. We also obtain formulas for the expectations and the higher moments of the Casson invariant and the order-3 knot invariant v3. These are the first precise formulas given for the distributions and higher moments of invariants in any model for random knots or links. We also use numerical computation to compare these to other random knot and link models, such as those based on grid diagrams. MSC: 57M25¨60B05 IntroductionIn this paper we study the distribution of finite type invariants of random knots and links. Our purpose is to investigate properties of typical knots, avoiding biases caused by focusing attention on a limited set of commonly studied examples. While tables of knots with up to 16 crossings have been compiled [18], and much is understood about infinite classes of knots, such as torus and alternating knots, we suspect that our view of the collection of all knots is distorted by the choices that simplicity and availability have given us. We have little knowledge of the distribution of knot invariants such as the Jones polynomial, or the linking number, among highly complicated knots and links. Studying a model of random knots allows us to probe for typical behavior beyond the familiar classes. As we elaborate below, the spectacular success of the probabilistic method in combinatorics makes us hopeful that it has much to offer in topology as well.A variety of models for random knots and links have been studied by physicists and biologists, as well as mathematicians. Common models are based on random 4-valent planar graphs with randomly assigned crossings, random diagrams on the integer grid in R 2 , Gaussian random polygons [10,5,29], and random walks on lattices in R 3 [35,33]. While many interesting numerical studies have been performed, and interesting results obtained in these models, there have been few rigorous derivations of associated statistical measures.In this paper we study a model of random knots and links called the Petaluma model, based on the representation of knots and links as petal diagrams that was introduced by Adams and studied in [2]. The Petaluma model has the advantage of being both universal, in that it represents all knots and links, and combinatorially simple, so that knots have simple descriptions in terms of a single permutation. We obtain here what appears to be the first precise formulas in any random model for the distributions of knot and link invariants.

The writhe of permutations and random framed knots

“…Our methods extend previous work on invariants of random knots [EZHLN16], and work by Mingo and Nica [MN98].…”

“…Now inv G is asymptotically logistic with density f (x) = 1 2 sech 2 x. See [EZHLN16] and Proposition 5 below.…”

“…A model for random framed knots is described, which refines the Petaluma model, studied in [EZHLN16]. The distribution of the framing in this model is equivalent to the writhe of random permutations.…”

“…In [EZHLN16], we study a model for random knots, in which a permutation π ∈ S 2n+1 defines a knot, through a petal diagram, from [ACD + 15]. In Section 2, we refine this random model to deal with framed knots, and observe that the knot's framing is equal to the writhe of π.…”

Random knots using Chebyshev billiard table diagrams

“…For this reason, we are interested in constructing a random model based on a particular knot diagram. The first paper on this topic appeared in 2014 by Even-Zohar, Hass, Linial, and Nowik [EZHLN14] using theübercrossing and petal projections of Adams et al [ACD + 12]. Forthcoming work of Dunfield, Obeidin, et al [Dun14] is expected to be on the topic of random knot diagrams as well.…”