A quantum combinatorial approach for computing a tetrahedral network of Jones-Wenzl projectors

TL;DR

This paper introduces a quantum combinatorial method for evaluating tetrahedral networks of Jones-Wenzl projectors, connecting graph evaluations to quantum algebra and Stirling numbers.

Contribution

It presents a novel quantum combinatorial approach for computing tetrahedral networks, linking graph invariants to Stirling numbers and quantized factorials.

Findings

Recovered two definitions of unsigned Stirling numbers of the first kind.

Established an equality for the quantized factorial using Stirling numbers.

Provided a new method for evaluating trivalent plane graphs in quantum topology.

Abstract

Trivalent plane graphs are used in various areas of mathematics which relate for instance to the colored Jones polynomial, invariants of 3-manifolds and quantum computation. Their evaluation is based on computations in the Temperley-Lieb algebra and more specifically the Jones-Wenzl projectors. We use the work by Kauffman-Lins to present a quantum combinatorial approach for evaluating a tetrahedral net. On the way we recover two equivalent definitions for the unsigned Stirling numbers of the first kind and we provide an equality for the quantized factorial using these numbers.