Carter, J. Scott; Lins, Sóstenes Thin-\(G\) theory and local moves for gems. (English) Zbl 0930.57018 Adv. Math. 143, No. 2, 251-283 (1999). Summary: Using the thin presentation of an \(n\)-gem, for \(n=3,4\), we prove that a set of local moves for crystallizations are sufficient in the sense that any two crystallizations inducing the same \(n\)-manifold are related by a finite sequence of moves taken from this set. In dimension 3, we apply the moves to prove an identity among quantum \(6j\)-symbols, and we propose a new model for a quantum topological invariant. In dimension 4, we observe that the thin presentation reduces the description of a 4-manifold to a 3-dimensional one. \(\copyright\) Academic Press. MSC: 57N10 Topology of general \(3\)-manifolds (MSC2010) 81T99 Quantum field theory; related classical field theories 57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) Keywords:quantum \(6j\)-symbols; local moves for crystallizations PDFBibTeX XMLCite \textit{J. S. Carter} and \textit{S. Lins}, Adv. Math. 143, No. 2, 251--283 (1999; Zbl 0930.57018) Full Text: DOI References: [1] J. S. Carter, D. E. Flath, M. Saito, The Classical and Quantum \(6j\); J. S. Carter, D. E. Flath, M. Saito, The Classical and Quantum \(6j\) [2] J. S. Carter, L. H. Kauffman, M. Saito, Singularities, diagrammatics, and their algebraic interpretations, Conference Proceedings of Top X, São Carlos, Brasil, 1996; J. S. Carter, L. H. Kauffman, M. Saito, Singularities, diagrammatics, and their algebraic interpretations, Conference Proceedings of Top X, São Carlos, Brasil, 1996 [3] Ferri, M., Colour switching and homemorphisms of manifolds, Canad. J. Math., 39, 8-32 (1987) · Zbl 0607.57014 [4] Ferri, M.; Gagliardi, C., Crystallization moves, Pacific J. Math., 100, 85-103 (1982) · Zbl 0517.57003 [5] Jones, V. F.R., A polynomial invariant for knots and links via von Neumann algebras, Bull. Amer. Math. Soc., 12, 103-111 (1985) · Zbl 0564.57006 [6] Kauffman, L. H., Knots and Physics (1991), World Scientific: World Scientific Singapore · Zbl 0749.57002 [7] Kauffman, L. H.; Lins, S. L., Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds (1994), Princeton Univ. Press: Princeton Univ. Press Princeton · Zbl 0821.57003 [8] Kirillov, A. N.; Reshetikhin, N. Yu., Representations of the algebra \(U_q ( sl q\), New Developments in the Theory of Knots (1989), World Scientific: World Scientific Singapore · Zbl 0742.17018 [9] Lawrence, R., Algebra and triangle relations, J. Pure Appl. Algebra, 100 (1995) · Zbl 0827.57011 [10] S. Lins, Twisors: Bridges among 3-manifolds, Discrete Math.; S. Lins, Twisors: Bridges among 3-manifolds, Discrete Math. · Zbl 0886.57010 [11] Lins, S., Invariant groups on equivalent crystallizations, European J. Combin., 10, 575-584 (1989) · Zbl 0688.57012 [12] Reshetikhin, N.; Turaev, V., Invariants of 3-manifolds via link polynomials, Invent. Math., 103, 547-597 (1991) · Zbl 0725.57007 [13] Turaev, V., Quantum Invariants of Knots and 3-Manifolds (1995), de Gruyter: de Gruyter Berlin · Zbl 0812.57003 [14] Turaev, V.; Viro, O. Ya., State sum invariants of 3-manifolds and quantum \(6j\), Topology, 31, 865-902 (1992) · Zbl 0779.57009 [15] Witten, E., Quantum field theory and the Jones polynomial, Comm. Math. Phys., 121, 715-750 (1989) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.