2001/11/06 by J. L. Jacobsen, Jacobsen, J. L., P. Zinn-Justin +1
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.math-ph/0111011
proceedings European Summer School St-Petersburg 2001
arxiv created 2001/11/06 · arxiv updated 2009/11/30
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 22 crossings. We then comment on their large-order asymptotic behavior.