The Combinatorics of Alternating Tangles: from theory to computerized enumeration

We study the enumeration of alternating links and tangles, considered up to topological (flype) equivalences. A weight $n$ is given to each connected component, and in particular the limit $n\to 0$ yields information about (alternating) knots. Using a finite renormalization scheme for an associated matrix model, we first reduce the task to that of enumerating planar tetravalent diagrams with two types of vertices (self-intersections and tangencies), where now the subtle issue of topological equivalences has been eliminated. The number of such diagrams with $p$ vertices scales as $12^p$ for $p\to\infty$. We next show how to efficiently enumerate these diagrams (in time $\sim 2.7^p$) by using a transfer matrix method. We give results for various generating functions up to 22 crossings. We then comment on their large-order asymptotic behavior.Comment: proceedings European Summer School St-Petersburg 200

La théorie de l'ordre de Poinsot à Bourgoin : Mathématiques, philosophie, art ornemental

International audienceThe aim of this paper is to understand the dynamics of the theory of order in the nineteenth century and to reveal a specific approach to mathematics, science, philosophy and decorative art in which order plays a prominent role. We will analyze the singular meaning that Poinsot assigns to the notion of order in the mathematical sciences, before describing the circulation of his writings on the order in the nineteenth century. Poinsot is one of the main sources of Cournot, who places the notions of order and form as the basis of his knowledge system. Then we will study the writings of Bourgoin who develops a combinatorics of ornaments based on the categories of order and form.L’enjeu de cet article est d’appréhender la dynamique de la théorie de l’ordre au xix e siècle et de mettre au jour une approche spécifique des mathématiques, des sciences, de la philosophie et de l’art ornemental dans laquelle l’ordre joue un rôle prééminent. Nous reviendrons sur la signification singulière que Poinsot assigne à l’ordre dans les sciences mathématiques, avant de décrire la circulation de ses écrits sur l’ordre au xix e siècle. Poinsot constitue l’une des principales sources de Cournot qui situe l’ordre et la forme au fondement de son système de connaissances. Enfin, nous nous intéresserons aux écrits de Bourgoin qui développe une combinatoire des ornements, fondée sur les catégories d’ordre et de forme