On a Background of the Existence of Multi-variable Link Invariants

• Journal title : Kyungpook mathematical journal
• Volume 48, Issue 2,  2008, pp.233-240
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2008.48.2.233
Title & Authors
On a Background of the Existence of Multi-variable Link Invariants
Nagasato, Fumikazu; Hamai, Kanau;

Abstract
We present a quantum theorical background of the existence of multi-variable link invariants, for example the Kauffman polynomial, by observing the quantum (sl(2,$\small{\mathbb{C}}$), ad)-invariant from the Kontsevich invariant point of view. The background implies that the Kauffman polynomial can be studied by using the sl(N,$\small{\mathbb{C}}$)-skein theory similar to the Jones polynomial and the HOMFLY polynomial.
Keywords
Kauffman polynomial;$\small{\Lambda}$-polynomial;Kontsevich invariant Q-invariant;weight system;
Language
English
Cited by
References
1.
R. D. Brandt, W. B. R. Lickorich and K. C. Millett, A polynomial invariant for unoriented knots and links, Invent. Math., 84(1986), 563-574.

2.
S. V. Chmutov and A. N. Varchenko, Remarks on the Vassiliev knot invariants coming from sl2, Topology, 36(1997), 153-178.

3.
K. Hamai, A combinatorial reconstruction of the quantum (sl(2,C),ad)-invariant through the Kontsevich invariant (in Japanese, joint work with F. Nagasato), Master thesis, Kyushu University (2002).

4.
C. F. Ho, A new polynomial for knots and links-preliminary report, Abstracts Amer. Math. Soc., 6(1985), 300.

5.
T. Kanenobu, Tangle surgeries on the double of a tangle and link polynomials, Kobe J. Math., 19(2002), 1-19.

6.
C. Kassel, Quantum Groups, GTM 155, Springer-Verlag, 1995.

7.
L. H. Kauffman, On knots, Ann. of Math. Studies, 115, Princeton University Press, 1987.

8.
T. T. Q. Le and J. Murakami, Kontsevich's integral for the Kauffman polynomial, Nagoya Math. J., 142(1996), 39-65.

9.
T. T. Q. Le and J. Murakami, The universal Vassiliev-Kontsevich invariant for framed oriented links, Compositio Mathematica, 102(1996), 41-64.

10.
F. Nagasato, A diagrammatic construction of the (sl(N,C), $\rho$)-weight system, Inter-disciplinary Information Sciences, 9(2003), 43-51.

11.
T. Ohtsuki, Quantum Invariants, Series on Knots and Everything, 29, World Scientific, 2002.

12.
Y. Yokota, Skeins and quantum SU(N) invariants of 3-manifolds, Math. Ann., 307(1997), 109-138.