Quantum relatives of the Alexander polynomial

Viro, O.

doi:10.1090/S1061-0022-07-00956-9

Quantum relatives of the Alexander polynomial
HTML articles powered by AMS MathViewer

by O. Viro
Translated by: the author

St. Petersburg Math. J. 18 (2007), 391-457

DOI: https://doi.org/10.1090/S1061-0022-07-00956-9

Published electronically: April 11, 2007

PDF | Request permission

Abstract:

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 $\operatorname {gl}(1|1)$ and $\operatorname {sl}(2)$. The corresponding face state sum models for the generalized Conway function are presented.

References

J. W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928), no. 2, 275–306. MR 1501429, DOI 10.1090/S0002-9947-1928-1501429-1
C. Blanchet, N. Habegger, G. Masbaum, and P. Vogel, Three-manifold invariants derived from the Kauffman bracket, Topology 31 (1992), no. 4, 685–699. MR 1191373, DOI 10.1016/0040-9383(92)90002-Y
J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) Pergamon, Oxford, 1970, pp. 329–358. MR 0258014
Tetsuo Deguchi and Yasuhiro Akutsu, Colored vertex models, colored IRF models and invariants of trivalent colored graphs, J. Phys. Soc. Japan 62 (1993), no. 1, 19–35. MR 1206148, DOI 10.1143/JPSJ.62.19
R. Fintushel and R. Stern, Knots, links, and \rom4-manifolds, Preprint, Differential Geometry dg-ga/9612014, 1996.
Louis H. Kauffman, Map coloring, $q$-deformed spin networks, and Turaev-Viro invariants for $3$-manifolds, Internat. J. Modern Phys. B 6 (1992), no. 11-12, 1765–1794. Topological and quantum group methods in field theory and condensed matter physics. MR 1186843, DOI 10.1142/S0217979292000852
A. N. Kirillov and N. Yu. Reshetikhin, Representations of the algebra ${U}_q(\textrm {sl}(2)),\;q$-orthogonal polynomials and invariants of links, Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988) Adv. Ser. Math. Phys., vol. 7, World Sci. Publ., Teaneck, NJ, 1989, pp. 285–339. MR 1026957
L. H. Kauffman and H. Saleur, Free fermions and the Alexander-Conway polynomial, Comm. Math. Phys. 141 (1991), no. 2, 293–327. MR 1133269
P. P. Kulish, Quantum Lie superalgebras and supergroups, Problems of modern quantum field theory (Alushta, 1989) Res. Rep. Phys., Springer, Berlin, 1989, pp. 14–21. MR 1091758
Shahn Majid and M. J. Rodríguez-Plaza, Nonstandard quantum groups and superization, J. Math. Phys. 36 (1995), no. 12, 7081–7097. MR 1359681, DOI 10.1063/1.531344
Jun Murakami, A state model for the multivariable Alexander polynomial, Pacific J. Math. 157 (1993), no. 1, 109–135. MR 1197048
Jun Murakami, The multi-variable Alexander polynomial and a one-parameter family of representations of $\scr U_q(\mathfrak {s}\mathfrak {l}(2,\textbf {C}))$ at $q^2=-1$, Quantum groups (Leningrad, 1990) Lecture Notes in Math., vol. 1510, Springer, Berlin, 1992, pp. 350–353. MR 1183500, DOI 10.1007/BFb0101201
L. Rozansky and H. Saleur, Quantum field theory for the multi-variable Alexander-Conway polynomial, Nuclear Phys. B 376 (1992), no. 3, 461–509. MR 1170953, DOI 10.1016/0550-3213(92)90118-U
L. Rozansky and H. Saleur, $S$- and $T$-matrices for the super $\textrm {U}(1,1)\ \textrm {WZW}$ model. Application to surgery and $3$-manifolds invariants based on the Alexander-Conway polynomial, Nuclear Phys. B 389 (1993), no. 2, 365–423. MR 1201534, DOI 10.1016/0550-3213(93)90326-K
N. Reshetikhin, Quantum supergroups, Quantum field theory, statistical mechanics, quantum groups and topology (Coral Gables, FL, 1991) World Sci. Publ., River Edge, NJ, 1992, pp. 264–282. MR 1223142
N. Yu. Reshetikhin and V. G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990), no. 1, 1–26. MR 1036112
N. Reshetikhin and V. G. Turaev, Invariants of $3$-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), no. 3, 547–597. MR 1091619, DOI 10.1007/BF01239527
V. G. Turaev, Reidemeister torsion in knot theory, Uspekhi Mat. Nauk 41 (1986), no. 1(247), 97–147, 240 (Russian). MR 832411
—, Topology of shadows, Preprint, 1991.
V. G. Turaev, Quantum invariants of knots and 3-manifolds, De Gruyter Studies in Mathematics, vol. 18, Walter de Gruyter & Co., Berlin, 1994. MR 1292673
V. G. Turaev and O. Ya. Viro, State sum invariants of $3$-manifolds and quantum $6j$-symbols, Topology 31 (1992), no. 4, 865–902. MR 1191386, DOI 10.1016/0040-9383(92)90015-A