1206.3457 (Hans-Christian Ruiz)

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

Hans-Christian Ruiz

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.

View original: http://arxiv.org/abs/1206.3457