Introduction to Spin Networks and Towards a Generalization of the Decomposition Theorem

TL;DR

This paper provides an accessible introduction to spin networks, their mathematical framework, and their relation to 3-manifold invariants, while also exploring a potential generalization of the decomposition theorem for non-planar networks.

Contribution

It offers a comprehensive overview of spin networks and invariants, and presents initial results towards a generalized decomposition theorem for non-planar spin networks.

Findings

Relation of spin networks to Ponzano-Regge theory

Construction of Turaev-Viro invariant from spin networks

Initial results on decomposition theorem for non-planar networks

Abstract

The objective of this work is twofold. On one hand, it is intended as a short introduction to spin networks and invariants of 3-manifolds. It covers the main areas needed to have a first understanding of the topics involved in the development of spin networks, which are described in a detailed but not exhaustive manner and in order of their conceptual development such that the reader is able to use this work as a first reading. A motivation due to R. Penrose for considering spin networks as a way of constructing a 3-D Euclidean space is presented, as well as their relation to Ponzano-Regge theory. Furthermore, the basic mathematical framework for the algebraic description of spin networks via quantum groups is described and the notion of a spherical category and its correspondence to the diagrammatic representation given by the Temperley-Lieb recoupling theory are presented. In order to give an example of topological invariants and their relation to TQFT the construction of the Turaev-Viro invariant is depicted and related to the Kauffman-Lins invariant. On the other hand, some results aiming at a decomposition theorem for non-planar spin networks are presented. For this, Moussouris' algorithm and some basic concepts of topological graph theory are explained and used, especially Kuratowski's theorem and the Rotation Scheme theorem.