We study the enumeration of alternating links and tangles, considered up to topo-logical (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 topo-logical equivalences has been eliminated. The number of such diagrams with p vertices scales as 12~p for p → ∞. We next show how to efficiently enumerate these diagrams (in time ～ 2.7~p) by using a transfer matrix method. We have obtained results for various generating functions up to 22 crossings. We then comment on their large-order asymptotic behavior.
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