AM  Vol.5 No.1 , January 2014
The Quantum sl2-Invariant of a Family of Knots
Abstract: We give a general formula of the quantum sl2-invariant of a family of braid knots. To compute the quantum invariant of the links we use the Lie algebra g=sl2 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 sl2-Invariant of a Family of Knots," Applied Mathematics, Vol. 5 No. 1, 2014, pp. 70-78. doi: 10.4236/am.2014.51008.

[1]   E. Witten, “Quantum Field Theory and the Jones Polynomial,” Communications in Mathematical Physics, Vol. 121, No. 3, 1989, pp. 351-399.

[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.

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

[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.

[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.

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

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

[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)