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The Combinatorics of Alternating Tangles: from theory to computerized enumeration

J. L. Jacobsen·P. Zinn-Justin
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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.

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J. L. Jacobsen, P. Zinn-Justin. 2001-11-06. The Combinatorics of Alternating Tangles: from theory to computerized enumeration. https://arxiv.org/abs/math-ph/0111011

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