Tabulating knot polynomials for arborescent knots

TL;DR

This paper develops a new Feynman diagram technique and a family approach to compute and tabulate colored knot polynomials for arborescent knots, enhancing the topological understanding and computational efficiency.

Contribution

It introduces a novel Feynman diagram method and auxiliary matrix model framework for calculating arborescent knot polynomials, enabling effective tabulation and analysis.

Findings

New tables of colored knot polynomials will be available online.

The family approach simplifies computations for arborescent knots.

Gauge invariance clarifies Racah matrix interpretations and sign conventions.

Abstract

Arborescent knots are the ones which can be represented in terms of double fat graphs or equivalently as tree Feynman diagrams. This is the class of knots for which the present knowledge is enough for lifting topological description to the level of effective analytical formulas. The paper describes the origin and structure of the new tables of colored knot polynomials, which will be posted at the dedicated site. Even if formal expressions are known in terms of modular transformation matrices, the computation in finite time requires additional ideas. We use the "family" approach, and apply it to arborescent knots in the Rolfsen table by developing a Feynman diagram technique associated with an auxiliary matrix model field theory. Gauge invariance in this theory helps to provide meaning to Racah matrices in the case of non-trivial multiplicities and explains the need for peculiar sign prescriptions in the calculation of [21]-colored HOMFLY polynomials.