On the Counting of Colored Tangles

TL;DR

This paper introduces matrix models that count colored alternating tangles, providing explicit generating functions and asymptotic analysis, connecting knot theory with statistical models and elliptic functions.

Contribution

It develops a novel matrix integral approach to count colored tangles, linking knot enumeration with elliptic functions and statistical models.

Findings

Generated explicit formulas for 2-color tangles

Expanded generating functions to 16 crossings

Analyzed asymptotic behavior of the models

Abstract

The connection between matrix integrals and links is used to define matrix models which count alternating tangles in 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 tangles is provided in terms of elliptic functions, expanded to 16-th order (16 crossings) and its asymptotic behavior is given.