Commutation classes of double wiring diagrams

TL;DR

This paper introduces a novel method for computing the graph of commutation classes of double wiring diagrams, enabling verification of a positivity conjecture for small cases.

Contribution

A new computational approach for analyzing commutation classes of double wiring diagrams and confirming a conjecture for small n.

Findings

Successfully computed the graph for up to five strings.

Confirmed the positivity conjecture for n less than five.

Provided a new method applicable to related combinatorial structures.

Abstract

We describe a new method for computing the graph of commutation classes of double wiring diagrams. Using these methods we compute the graph for five strings or less which allows us to confirm a positivity conjecture of Fomin and Zelevinsky when n is less than five .