Skein relations for spin networks, modified

TL;DR

This paper introduces a simplified diagrammatic framework for spin networks using a new sign convention, connecting tensor calculus with spin network evaluations and extending graph labelings to complex values.

Contribution

It proposes a new sign convention and background space for spin networks, simplifying recoupling rules and extending evaluations to complex labels.

Findings

Simplified recoupling rules with a new sign convention

Preservation of chromatic graph evaluation standards

Extension of graph labels to complex values

Abstract

An alternative framework underlying connection between tensor ${\rm sl}_2$-calculus and spin networks is suggested. New sign convention for the inner product in the dual spinor space leads to a simpler and direct set of initial rules for the diagrammatic recoupling methods. Yet it preserves the standard chromatic graph evaluations. In contrast with the standard formulation, the background space is that of symmetric tensor spaces, which seems to be in accordance with the representation theory of ${\rm SL}(2)$. An example of Apollonian disk packing is shown to be a source of spin networks. The graph labeling is extended to non-integer values, resulting in the complex-values of chromatic evaluations.