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 → 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 12p for p → ∞. We next show how to efficiently enumerate these diagrams (in time ~ 2.7p) by using a transfer matrix method. We give results for various generating functions up to 21 crossings. We then comment on their large-order asymptotic behavior.
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Jacobsen, J.L., Zinn-Justin, P. (2002). The Combinatorics of Alternating Tangles: From Theory to Computerized Enumeration. In: Malyshev, V., Vershik, A. (eds) Asymptotic Combinatorics with Application to Mathematical Physics. NATO Science Series, vol 77. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0575-3_5
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DOI: https://doi.org/10.1007/978-94-010-0575-3_5
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