On the counting of colored tangles

Abstract

The connection between matrix integrals and links is used to define matrix models which count alternating tangles n which each closed loop is weighted with a factor n, i.e. may be regarded as decorated with n possible colors. For n=2, the corresponding matrix integral is that recently solved in the study of the random lattice six-vertex model. The generating function of alternating 2-color tangle is provided in terms of elliptic functions, expanded to 16-th order (16 crossings) and its asymptotic behaviors is given.