The General O(n) Quartic Matrix Model and its application to Counting Tangles and Links

TL;DR

This paper introduces a quantum field-theoretic matrix model approach to count alternating tangles and links, removing overcounting via renormalization, and provides exact solutions for specific cases.

Contribution

It presents a unified matrix model framework with renormalization to accurately count tangles and links, advancing topological enumeration methods.

Findings

Derived renormalization equations for the matrix model

Solved specific cases exactly

Established a unified counting framework for tangles and links

Abstract

The counting of alternating tangles in terms of their crossing number, number of external legs and connected components is presented here in a unified framework using quantum field-theoretic methods applied to a matrix model of colored links. The overcounting related to topological equivalence of diagrams is removed by means of a renormalization scheme of the matrix model; the corresponding ``renormalization equations'' are derived. Some particular cases are studied in detail and solved exactly.