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.

There is much recent interest in understanding the density at which constant size graphs can appear in a very large graph. Specifically, the inducibility of a graph H is its extremal density, as an induced subgraph of G, where |G| → ∞. Already for 4-vertex graphs many questions are still open. Thus, the inducibility of the 4-path was addressed in a construction of Exoo (Ars Combin 22:5-10, 1986), but remains unknown. Refuting a conjecture of Erdős, Thomason (Combinatorica 17(1):125-134, 1997) constructed graphs with a small density of both 4-cliques and 4-anticliques. In this note, we merge these two approaches and construct better graphs for both problems.

Let A and B be two affinely generating sets of Z n 2 . As usual, we denote their Minkowski sum by A + B. How small can A + B be, given the cardinalities of A and B? We give a tight answer to this question. Our bound is attained when both A and B are unions of cosets of a certain subgroup of Z n 2 . These cosets are arranged as Hamming balls, the smaller of which has radius 1.

The study of knots and links from a probabilistic viewpoint provides insight into the behavior of ''typical'' knots, and opens avenues for new constructions of knots and other topological objects with interesting properties. The knotting of random curves arises also in applications to the natural sciences, such as in the context of the structure of polymers. We present here several known and new randomized models of knots and links. We review the main known results on the knot distribution in each model. We discuss the nature of these models and the properties of the knots they produce. Of particular interest to us are finite type invariants of random knots, and the recently studied Petaluma model. We report on rigorous results and numerical experiments concerning the asymptotic distribution of such knot invariants. Our approach raises questions of universality and classification of the various random knot models.

The representation of knots by petal diagrams (Adams et al. 2012) naturally defines a sequence of distributions on the set of knots. In this article we establish some basic properties of this randomized knot model. We prove that in the random n-petal model the probability of obtaining every specific knot type decays to zero as n, the number of petals, grows. In addition we improve the bounds relating the crossing number and the petal number of a knot. This implies that the n-petal model represents at least exponentially many distinct knots.Past approaches to showing, in some random models, that individual knot types occur with vanishing probability, rely on the prevalence of localized connect summands as the complexity of the knot increases. However this phenomenon is not clear in other models, including petal diagrams, random grid diagrams, and uniform random polygons. Thus we provide a new approach to investigate this question.MSC 57M25, 60B05