A transfer matrix approach to the enumeration of colored links

TL;DR

This paper introduces a transfer matrix algorithm for enumerating colored link diagrams, incorporating topological equivalences and analyzing asymptotic behaviors, with explicit polynomial results up to certain complexities.

Contribution

It presents a novel transfer matrix approach for counting colored link diagrams considering topological equivalences and provides explicit polynomial enumerations.

Findings

Explicit polynomial generating functions up to order 19 for link diagrams

Solution of the large $n$ limit explicitly

Asymptotic analysis matches KPZ conjecture for $0 o 2$

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 (link diagrams), 15 (prime alternating tangles) and 11 (6-legged links) intersections. The limit $n\to\infty$ is solved explicitly. We then analyze the large-order asymptotics of the generating functions. For $0\le n \le 2$ good agreement is found with a conjecture for the critical exponent, based on the KPZ relation.