The Combinatorics of Alternating Tangles: from theory to computerized enumeration

TL;DR

This paper develops a method to enumerate alternating links and tangles using matrix models and transfer matrix techniques, providing new insights into their combinatorial structure and asymptotic behavior.

Contribution

It introduces a finite renormalization scheme to reduce the enumeration of topologically equivalent diagrams to counting planar tetravalent diagrams, enabling efficient computation.

Findings

Number of diagrams with p vertices scales as 12^p

Enumeration method runs in time ~2.7^p

Generated data for diagrams up to 22 crossings

Abstract

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.