The Quantum sl_{2}-Invariant of a Family of Knots

ABSTRACT

We
give a general formula of the quantum sl_{2}-invariant
of a family of braid knots. To compute the quantum invariant of the links we
use the Lie algebra g=sl_{2} in its standard two-dimensional
representation. We also recover the Jones polynomial of these knots as a
special case of this quantum invariant.

Cite this paper

A. Nizami, M. Munir and M. Bano, "The Quantum sl_{2}-Invariant of a Family of Knots," *Applied Mathematics*, Vol. 5 No. 1, 2014, pp. 70-78. doi: 10.4236/am.2014.51008.

A. Nizami, M. Munir and M. Bano, "The Quantum sl

References

[1] E. Witten, “Quantum Field Theory and the Jones Polynomial,” Communications in Mathematical Physics, Vol. 121, No. 3, 1989, pp. 351-399. http://dx.doi.org/10.1007/BF01217730

[2] N. Reshetikhin and V. Turaev, “Ribbon Graphs and Their Invariants Derived from Quantum Groups,” Communications in Mathematical Physics, Vol. 127, No. 1, 1990, pp. 1-26.

http://dx.doi.org/10.1007/BF02096491

[3] V. Turaev, “The Yang-Baxter Equation and Invariants of Links,” Inventiones Mathematicae, Vol. 92, No. 3, 1988, pp. 527-553.

http://dx.doi.org/10.1007/BF01393746

[4] V. G. Drinfeld, “Hopf Algebras and the Quantum Yang-Baxter Equation,” Soviet Mathematics Doklady, Vol. 32, 1985, pp. 254-258.

[5] V. G. Drinfeld, “Quantum Groups,” Proceedings of the International Congress of Mathematicians (Berkely, 1986),” American Mathematical Society, Providence, 1987, pp. 798-820.

[6] M. Jimbo, “A q-Difference Analogue of U(g) and the Yang-Baxter Equation,” Letters in Mathematical Physics, Vol. 10, No. 1, 1985, pp. 63-69. http://dx.doi.org/10.1007/BF00704588

[7] V. F. R. Jones, “A Polynomial Invariant for Links via Neumann Algebras,” Bulletin of the AMS—American Mathematical Society, Vol. 129, 1985, pp. 103-112. http://dx.doi.org/10.1090/S0273-0979-1985-15304-2

[8] L. H. Kauffman, “State models and Jones Polynomial,” Topology, Vol. 26, No. 3, 1987, pp. 395-407.

http://dx.doi.org/10.1016/0040-9383(87)90009-7

[9] V. F. R. Jones, “The Jones Polynomial,” Discrete Mathematics, Vol. 294, No. 3, 2005, pp. 275-277.

http://dx.doi.org/10.1016/j.disc.2004.10.024

[10] T. Ohtsuki, “Quantum Invariants: A Study of Knots, 3-Manifolds and Their Sets,” World Scientific, Sigapore City, 2002.

[11] S. Chmutov, S. Duzhin and J. Mostovoy, “Introduction to Vassiliev Knot Invariants,” 2011. (A Preliminary Draft Version of a Book on Vassiliev Knot Invariants)

[1] E. Witten, “Quantum Field Theory and the Jones Polynomial,” Communications in Mathematical Physics, Vol. 121, No. 3, 1989, pp. 351-399. http://dx.doi.org/10.1007/BF01217730

[2] N. Reshetikhin and V. Turaev, “Ribbon Graphs and Their Invariants Derived from Quantum Groups,” Communications in Mathematical Physics, Vol. 127, No. 1, 1990, pp. 1-26.

http://dx.doi.org/10.1007/BF02096491

[3] V. Turaev, “The Yang-Baxter Equation and Invariants of Links,” Inventiones Mathematicae, Vol. 92, No. 3, 1988, pp. 527-553.

http://dx.doi.org/10.1007/BF01393746

[4] V. G. Drinfeld, “Hopf Algebras and the Quantum Yang-Baxter Equation,” Soviet Mathematics Doklady, Vol. 32, 1985, pp. 254-258.

[5] V. G. Drinfeld, “Quantum Groups,” Proceedings of the International Congress of Mathematicians (Berkely, 1986),” American Mathematical Society, Providence, 1987, pp. 798-820.

[6] M. Jimbo, “A q-Difference Analogue of U(g) and the Yang-Baxter Equation,” Letters in Mathematical Physics, Vol. 10, No. 1, 1985, pp. 63-69. http://dx.doi.org/10.1007/BF00704588

[7] V. F. R. Jones, “A Polynomial Invariant for Links via Neumann Algebras,” Bulletin of the AMS—American Mathematical Society, Vol. 129, 1985, pp. 103-112. http://dx.doi.org/10.1090/S0273-0979-1985-15304-2

[8] L. H. Kauffman, “State models and Jones Polynomial,” Topology, Vol. 26, No. 3, 1987, pp. 395-407.

http://dx.doi.org/10.1016/0040-9383(87)90009-7

[9] V. F. R. Jones, “The Jones Polynomial,” Discrete Mathematics, Vol. 294, No. 3, 2005, pp. 275-277.

http://dx.doi.org/10.1016/j.disc.2004.10.024

[10] T. Ohtsuki, “Quantum Invariants: A Study of Knots, 3-Manifolds and Their Sets,” World Scientific, Sigapore City, 2002.

[11] S. Chmutov, S. Duzhin and J. Mostovoy, “Introduction to Vassiliev Knot Invariants,” 2011. (A Preliminary Draft Version of a Book on Vassiliev Knot Invariants)