Knotted 4-regular graphs. II: Consistent application of the Pachner moves. (English) Zbl 1543.83024
Summary: A common choice for the evolution of the knotted graphs in loop quantum gravity is to use the Pachner moves, adapted to graphs from their dual triangulations. Here, we show that the natural way to consistently use these moves is on framed graphs with edge twists, where the Pachner moves can only be performed when the twists, and the vertices the edges are incident on, meet certain criteria. For other twists, one can introduce an algebraic object, which allow any knotted graph with framed edges to be written in terms of a generalized braid group.
©2024 American Institute of Physics
For Part I, see [the author, J. Math. Phys. 63, No. 6, Article ID 063502, 19 p. (2022; Zbl 1508.57030)]
©2024 American Institute of Physics
For Part I, see [the author, J. Math. Phys. 63, No. 6, Article ID 063502, 19 p. (2022; Zbl 1508.57030)]
MSC:
| 83C45 | Quantization of the gravitational field |
| 57M15 | Relations of low-dimensional topology with graph theory |
| 20F36 | Braid groups; Artin groups |
Citations:
Zbl 1508.57030Software:
ReginaReferences:
| [1] | Alexander, J. W., A lemma on systems of knotted curves, Proc. Natl. Acad. Sci. U. S. A., 9, 93, 1923 · JFM 49.0408.03 · doi:10.1073/pnas.9.3.93 |
| [2] | Bilson-Thompson, S.; Hackett, J.; Kauffman, L.; Wan, Y., Emergent braided matter of quantum gravity, SIGMA, 8, 014, 2012 · Zbl 1242.83034 · doi:10.3842/SIGMA.2012.014 |
| [3] | Burton, B. A.; Budney, R.; Pettersson, W., Regina: Software for low-dimensional topology, 19992023 |
| [4] | Burton, B. A., Simplification paths in the Pachner graphs of closed orientable 3-manifold triangulations |
| [5] | Burton, B. A.; Pettersson, W., An edge-based framework for enumerating 3-manifold triangulations, 270-284, 2015, Schloss Dagstuhl - Leibniz-Zentrum für Informatik · Zbl 1378.68161 |
| [6] | Cartin, D., Knotted 4-regular graphs: Polynomial invariants and the Pachner moves, J. Math. Phys., 63, 063502, 2022 · Zbl 1508.57030 · doi:10.1063/5.0088228 |
| [7] | Elhamdadi, M.; Hajij, M.; Istvan, K., Framed knots, Math. Intel., 42, 7-22, 2020 · Zbl 1462.57003 · doi:10.1007/s00283-020-09990-0 |
| [8] | Markopoulou, F.; Prémont-Schwarz, I., Conserved topological defects in non-embedded graphs in quantum gravity, Classical Quantum Gravity, 25, 205015, 2008 · Zbl 1152.83014 · doi:10.1088/0264-9381/25/20/205015 |
| [9] | Pachner, U., P.L. homeomorphic manifolds are equivalent by elementary shellings, Eur. J. Combinatorics, 12, 129-145, 1991 · Zbl 0729.52003 · doi:10.1016/s0195-6698(13)80080-7 |
| [10] | Smolin, L.; Wan, Y., Propagation and interaction of chiral states in quantum gravity, Nucl. Phys. B, 796, 331-359, 2008 · Zbl 1219.83080 · doi:10.1016/j.nuclphysb.2007.12.018 |
| [11] | Wan, Y., On braid excitations in quantum gravity |
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