A Transfer Matrix approach to the Enumeration of Colored Links

Abstract

We propose a transfer matrix algorithm for the enumeration of alternating link diagrams with external legs, giving a weight n to each connected component. Considering more general tetravalent diagrams with self-intersections and tangencies allows us to treat topological (flype) equivalences. This is done by means of a finite renormalization scheme for an associated matrix model. We give results, expressed as polynomials in n, for the various generating functions up to order 19 (2-legged link diagrams), 15 (tangles) and 11 (6-legged links) crossings. The limit n→∞ is solved explicitly. We then analyze the large-order asymptotics of the generating functions. For 0≤n≤2 good agreement is found w..