Two variable link polynomials from quantum supergroups. (English) Zbl 0777.57005
In this letter is observed that quantum supergroups offer new insight into the construction of two-variable link polynomials through the use of non-trivial one-parameter families of representations. New two-variable link polynomials are constructed corresponding to such a family of representations of \(U_ q(\text{gl}(2| 1))\). Their connection with the Kauffman polynomials is also investigated.
Reviewer: J.R.Links
MSC:
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
| 81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |
References:
| [1] | Wadati, M., Deguchi, T., and Akutsu, Y., Phys. Rep. 180, 248 (1989); Akutsu, Y., Deguchi, T., and Wadati, M., in C. N. Yang and M. L. Ge (eds), Braid Group, Knot Theory and Statistical Mechanics, World Scientific, Singapore, 1989, p. 151. · Zbl 0847.57007 · doi:10.1016/0370-1573(89)90123-3 |
| [2] | Witten, E., Comm. Math. Phys. 121, 351 (1989); Nuclear Phys. B330, 285 (1990). · Zbl 0667.57005 · doi:10.1007/BF01217730 |
| [3] | Reshetikhin, N. Yu., Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links I, II, LOMI preprints E-4-87, E-17-87. |
| [4] | Zhang, R. B., Gould, M. D., and Bracken, A. J., Comm. Math. Phys. 137, 13 (1991). · Zbl 0735.17018 · doi:10.1007/BF02099115 |
| [5] | Kauffman, L. and Saleur, H., Free fermions and the Conway-Alexander polynomial, Comm. Math. Phys. (to appear); Fermions and link invariants, preprint YCTP-P21-91 (1991). · Zbl 0751.57004 |
| [6] | Deguchi, T. and Akutsu, Y., J. Phys. A. 23, 1861 (1990). · Zbl 0724.57002 · doi:10.1088/0305-4470/23/11/014 |
| [7] | Gould, M. D., Tsohantjis, I., and Bracken, A. J., Quantum supergroups and link polynomials, Rev. Math. Phys. (in press). · Zbl 0797.17008 |
| [8] | Links, J., Gould, M. D., and Zhang, R. B., University of Queensland preprint (1992) (to appear). |
| [9] | Lee, H. C., Invariants of quantum groups, link polynomials of U q (gl(N+1, C); L) and holonomy in SU(N+1?L, L) Chern-Simons theory, Chalk River preprint TP-90, 1205 (1991). |
| [10] | Khoroshkin, S. M. and Tolstoy, V. N., Comm. Math. Phys. 141, 599 (1991). · Zbl 0744.17015 · doi:10.1007/BF02102819 |
| [11] | Zhang, R. B. and Gould, M. D., J. Math. Phys. 32, 3261 (1991). · Zbl 0759.17011 · doi:10.1063/1.529487 |
| [12] | Jones, V. F. R., Bull. Amer. Math. Soc. 12, 103 (1985). · Zbl 0564.57006 · doi:10.1090/S0273-0979-1985-15304-2 |
| [13] | Freyed, P., Yetter, D., Hoste, J., Lickorish, W. B. R., Millet, K., and Ocneanu, A., Bull. Amer. Math. Soc. 12, 239 (1985). · Zbl 0572.57002 · doi:10.1090/S0273-0979-1985-15361-3 |
| [14] | Kauffman, L., Trans. Amer. Math. Soc. 318, 417 (1990). · Zbl 0763.57004 · doi:10.2307/2001315 |
| [15] | Gould, M. D. and Zhang, R. B., J. Math. Phys. 31, 1524 (1990). · Zbl 0717.17024 · doi:10.1063/1.528695 |
| [16] | Kac, V. G., Adv. Math. 26, 8 (1977). · Zbl 0366.17012 · doi:10.1016/0001-8708(77)90017-2 |
| [17] | Birman, J. and Wenzl, H., Trans. Amer. Math. Soc. 313, 249 (1989). · doi:10.1090/S0002-9947-1989-0992598-X |
| [18] | Murakami, J., Osaka J. Math. 24, 745 (1987). |
| [19] | Zhang, R. B., J. Math. Phys. 32, 2605 (1991). · Zbl 0751.17017 · doi:10.1063/1.529105 |
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