Surface embedding, topology and dualization for spin networks

TL;DR

This paper explores the embedding and topological properties of spin networks derived from angular momentum symbols, focusing on their surface embeddings, dualizations, and minimal genus representations.

Contribution

It introduces new insights into the surface embeddings and dualizations of spin networks, including minimal genus embeddings into various surfaces and their triangulations.

Findings

Identifies embeddings of spin networks into spheres, tori, projective spaces, and Klein bottles.

Shows dual 2-skeletons form triangulations of these surfaces.

Finds two families of 3nj graphs with minimal genus embeddings.

Abstract

Spin networks are graphs derived from 3nj symbols of angular momentum. The surface embedding, the topology and dualization of these networks are considered. Embeddings into compact surfaces include the orientable sphere S^2 and the torus T, and the not orientable projective space P^2 and Klein's bottle K. Two families of 3nj graphs admit embeddings of minimal genus into S^2 and P^2. Their dual 2-skeletons are shown to be triangulations of these surfaces.