On the counting of colored tangles (Q2706867)

The paper is a continuation of a previous work of the same authors [``Matrix integrals and the counting of tangles and links'', preprint, \url{http://arXiv.org/abs/math-ph/9904019}]. Using the method of matrix integrals and Feynman diagrams the authors are able to count 2-string alternating tangles with a fixed number of crossings (where the tangles are allowed to contain any number of closed curves). In this and the previous paper the authors reproduced results of \textit{C. Sundberg} and \textit{M. B. Thistlethwaite} [Pac. J. Math. 182, No. 2, 329-358 (1998; Zbl 0901.57008)] concerning the rate of growth of the number of prime alternating tangles. This counting is possible because of the Tait-conjecture that affirms that two such diagrams are equivalent iff they are ``flype-equivalent'', see the article by \textit{W. W. Menasco} and \textit{M. B. Thistlethwaite} [Bull. Am. Math. Soc., New Ser. 25, No. 2, 403-412 (1991; Zbl 0745.57002)]. The new work in this paper concerns the problem of counting alternating tangles with a fixed number \(n\) of closed loops. One can think of such tangles as decorated with \(n\) possible colors, one for each closed loop. The authors formulate this in terms of a matrix integral. Solving the matrix integral for any \(n\) remains out of reach for the moment. However, in the particular case \(n=2\) the model is solved and explicit numbers are given up to (including) 16 crossings. Finally the paper concludes with some statements and conjectures about the asymptotic behavior for a large number of crossings.