Skein relations for spin networks, modified
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- July 2018
- DOI:
- 10.48550/arXiv.1807.07244
- arXiv:
- arXiv:1807.07244
- Bibcode:
- 2018arXiv180707244K
- Keywords:
-
- Mathematical Physics;
- 20C35;
- 57M25;
- 57M27
- E-Print:
- 21 pages, lots of diagrams. Preprint of an article accepted for publication in Journal of Knot Theory and Its Ramifications [2018], World Scientific Publishing Company