Quantum relatives of the Alexander polynomial
Algebra i analiz, Tome 18 (2006) no. 3, pp. 63-157 Cet article a éte moissonné depuis la source Math-Net.Ru

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The multivariable Conway function is generalized to oriented framed trivalent graphs equipped with additional structure (coloring). This is done via refinements of Reshetikhin–Turaev functors based on irreducible representations of quantized $\mathrm{gl}(1|1)$ and $\mathrm{sl}(2)$. The corresponding face state sum models for the generalized Conway function are presented.
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O. Ya. Viro. Quantum relatives of the Alexander polynomial. Algebra i analiz, Tome 18 (2006) no. 3, pp. 63-157. http://geodesic.mathdoc.fr/item/AA_2006_18_3_a2/

[1] J. W. Alexander, “Topological invariants of knots i links”, Trans. Amer. Math. Soc., 30 (1928), 275–306 | DOI | MR | Zbl

[2] C. Blanchet, N. Habegger, G. Masbaum, P. Vogel, “Three-manifold invariants derived from the Kauffman bracket”, Topology, 31 (1992), 685–699 | DOI | MR | Zbl

[3] J. H. Conway, “An enumeration of knots i links, and some of their algebraic properties”, Computational problems in Abstract Algebra, ed. J. Leech, Pergamon Press, New York, 1970, 329–358 | MR

[4] T. Deguchi, Y. Akutsu, “Colored Vertex Models, Colored IRF Models and Invariants of Trivalent Colored Graphs”, J. Phys. Soc. Japan, 62 (1993), 19–35 ; T. Deguchi, Multivariable Invariants of Colored Links и Related Solvable Models in Statistical Mechanics, Thesis, University of Tokio, March 1992 | DOI | MR | Zbl

[5] R. Fintushel, R. Stern, Knots, Links, and 4-Manifolds, Preprint, Differential Geometry , 1996 dg-ga/9612014 | MR

[6] L. Kauffman, “Map Coloring, $q$-Deformed Spin-Networks, and the Turaev–Viro Invariants for 3-manifolds”, Int. J. of Modern Physics B, 6:11/12 (1992), 1765–1794 | DOI | MR | Zbl

[7] A. N. Kirillov, N. Y. Reshetikhin, “Representations of the algebra $U_q(sl_2)$, $q$-orthogonal polynomials and invariants of links”, Infinite dimensional Lie algebras and groups, Adv. Ser. in Math. Phys., 7, ed. V. G. Kac, World Scientific, Singapore, 1988, 285–339 | MR

[8] L. Kauffman, H. Saleur, “Free fermions and the Alexander-Conway polynomial”, Commun. Math. Phys., 141 (1991), 293–327 | DOI | MR | Zbl

[9] P. P. Kulish, “Quantum Lie superalgebras and supergroups”, Problems of Modern Quantum Field Theory, Proceedings of Alushta conference (May 1989), eds. Belavin A. A., Klimyk A. U., Zamolodchikov A. B., Springer-Verlag, 1989, 14–21 | MR

[10] Shahn Majid, M. J. Rodríguez-Plaza, Non-standard quantum groups and superization, Preprint , 13 June 1995 q-alg/9506015 | MR

[11] Jun Murakami, “A State Model for the Multi-Variable Alexander Polynomial”, Pacific J. of Math., 157 (1993), 109–135 | MR | Zbl

[12] Jun Murakami, “The Multi-Variable Alexander Polynomial and a One-Parameter Family of Representations of $\mathfrak{U}_q(\mathfrak{sl}(2,\mathbb C))$”, Quantum Groups (Leningrad, 1990), Lecture Notes in Math., 1510, 1992, 350–353 | MR | Zbl

[13] L. Rozansky, H. Saleur, “Quantum field theory for the multivariable Alexander-Conway polynomial”, Nuclear Physics B, 376 (1992), 461–509 | DOI | MR

[14] L. Rozansky, H. Saleur, “$S$- and $T$-matrices for the super $U(1,1)$ WZW model. Application to surgery and 3-manifolds invariants based on the Alexander-Conway polynomial”, Nuclear Physics B, 389 (1993), 365–423 | DOI | MR

[15] N. Yu. Reshetikhin, “Quantum Supergroups”, Quantum Field Theory Statistical Mechanics, Quantum Groups i Topology, Proceedings of the NATO advanced research workshop (Coral Gables, FL, 1991), World Sci. Publishing, River Edge, NJ, 1992, 264–282 | MR | Zbl

[16] N. Yu. Reshetikhin and V. G. Turaev, “Ribbon graphs and their invariants derived from quantum groups”, Commun. Math. Phys., 127 (1990), 1–26 | DOI | MR | Zbl

[17] N. Yu. Reshetikhin and V. G. Turaev, “Invariants of 3-manifolds via link polynomials and quantum groups”, Invent. Math., 103 (1991), 547–598 | DOI | MR

[18] Russian Math. Surveys, 41 (1986), 119–182 | DOI | MR | Zbl

[19] V. Turaev, Topology of shadows, Preprint, 1991

[20] V. Turaev, Quantum Invariants of Knots and 3-Manifolds, Walter de Gruyter, Berlin, New York, 1994 | MR | Zbl

[21] V. Turaev and O. Viro, “State sum invariants of 3-manifolds and quantum $6j$-symbols”, Topology, 31 (1992), 865–902 | DOI | MR | Zbl