Generalized Barrett-Crane Vertices and Invariants of Embedded Graphs

TL;DR

This paper introduces q-analogues of Barrett-Crane vertices using SU(2) recoupling theory, generalizing to unframed n-vertices and analyzing their invariants of embedded graphs.

Contribution

It develops a new class of q-analogues for Barrett-Crane vertices and extends the theory to unframed n-vertices with their invariants.

Findings

Defined q-analogues of 4-vertices in Spin(4) recoupling theory

Generalized to operators with unframed n-vertex symmetry

Examined properties of invariants of embedded unframed graphs

Abstract

We describe q-analogues of the 4-vertices in the Spin(4)-recoupling theory introduced by Barrett and Crane in gr-qc/9709028 using Kauffman-Lins SU(2)-recoupling theory in each factor and generalize them to obtain operators with the symmetry properties of unframed n-vertices. The elementary properties of the resulting invariants of embedded unframed graphs are examined.