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

Natural law, state formation and the foundations of social theory

This article proceeds from the claim that the earliest examples of sociological method were linked by a critique of the theories of natural law proposed in the Enlightenment. It illustrates this point by surveying the rise of social theory in the aftermath of the French Revolution and by examining approaches to natural law in key texts in the history of sociology. However, the article claims that natural law remains a blind spot for sociological method, and the original sociological dismissal of natural law as a formal body of normative postulates has prevented the formation of sociology as a comprehensive system of social interpretation. Using a series of historical examples, the article then argues that natural law theory needs in itself to be viewed sociologically, and, throughout modern history, ius-natural thinking has served not only to establish formal norms for evaluating social and political practices, but also to form the positive preconditions for the evolution of contemporary society in its distinctive differentiated form. The article concludes by arguing that a sociological reconstruction of natural law points the way towards a sociological theory of theory, which has typically proved elusive for sociological methodologies

Borders and Modal Articulations. Semiotic Constructs of Sensemaking Processes Enabling a Fecund Dialogue Between Cultural Psychology and Clinical Psychology

The notion of the border is an interesting advancement in research on the processes of meaning making within the cultural psychology. The development of this notion in semiotic key allows to handle with adequate complexity construction, transformation, stability and the breakup of the relationship between person/world/ otherness. These semiotic implications have already been widely discussed and ex- posed by authors such Valsiner (2007, 2014), Neuman (2003, 2008), Simão (Culture & Psychology, 9, 449–459, 2003, Theory & Psychology, 15, 549–574, 2005, 2015), with respect to issues of identity/relatedness, inside/outside, stability/change in the irrevers- ible flow of the time. In this work, after showing some of the basics of such semiotic notion of border, we discuss the processes of construction and transformation of borders through the modal articulation, defined as the contextual positioning that the person assumes with respect to the establishment of a boundary in terms of necessity, obligation, willingness, possibility, permission, ability. This modal subjective position- ing acquires considerable interest from the clinical point of view since its degree of plasticity vs that of rigidity is the basis of processes of development or stiffening of relations between person/world/otherness