Virtual braids, virtual Temperley-Lieb algebra and f-polynomial

Abstract

The authors study the properties of virtual Temperley-Lieb algebra and show how the f-polynomial of virtual knot can be derived from a representation of the virtual braid group into the virtual Temperley-Lieb algebra, which is an approach similar to Jones’s original construction.

Keywords

Virtual braids / Virtual Temperley-Lieb algebra / f-polynomial

References

[1] Dye, H. A. and Kauffman, L. H., Virtual Knot Diagrams and the Witten-Reshetikhin-Turaev Invariant, arXiv: math.GT/0407407.

[2] Goussarov M., Polyak M., Viro O.. Finite type invariants of classical and virtual knots. Topology, 2000, 39(5): 1045-1068

[3] Jones V. F. R.. A polynomial invariant for knots via von neuman algebras. Bull. Amer. Math. Soc., 1985, 12: 103-111

[4] Jones V. F. R.. Braid groups, Heck algebra and type II factors, Geometric Methods in Abstract Algebras. Proc. U.S.-Japan Symposium, 1986 242-273

[5] Jones V. F. R.. Heck algebra representations of braid groups and link polynomials. Ann. of Math., 1987, 126: 335-388

[6] Kamada S.. Braid presentation of virtual knots and welded knots. Osaka J. Math., 2007, 44(2): 441-458

[7] Kauffman L. H.. State models and the Jones polynomial. Topology, 1987, 26: 395-407

[8] Kauffman L. H.. An invariant of regular isotopy. Transactions of the American Mathematical Society, 1990, 318(2): 417-471

[9] Kauffman L. H.. Knots and Physics, 1991, Singapore: World Scientific

[10] Kauffman L. H.. Virtual knots, Talks at MSRI Meeting, January 1997 and AMS Meeting at University of Maryland, 1997

[11] Kauffman L. H.. Virtual knot theory. European J. Combin, 1999, 20: 663-690

[12] Kauffman L. H.. W M., M T.. Knot diagrammatics. Handbook of Knot Theory, 2005 233-318

[13] Kauffmann L. H., Lambropoulou S.. Virtual braids. Fund. Math., 2004, 184: 159-186

[14] Kauffmann L. H., Lambropoulou S.. Virtual braid and L-move. J. Knot Theory Ramifications, 2006, 15(6): 773-811

[15] Kauffman, L. H. and Lin, S. L., Temperley-Lieb Recoupling Theory and Invariants of Three-Manifold, Ann. of Math. Stud., 114, Princeton Univ. Press, Princeton, 1994.

[16] Krebes D. A.. Persistent invariants of tangles. J. Knot Theory Ramifications, 2000, 19(4): 471-477

[17] Manturov V. O.. Virtual knots: The State of Art, 2012

[18] Temperley H. N. V., Lieb E. H.. Relations between the percolation and colouring problem and other graph-theoretical problems associated with regular planar lattices: Some exact results for the percolation problem. Proc. Roy. Soc., A, 1971, 322: 251-280

[19] Vershinin V.. On homology of virtual braids and Burau representation. J. Knot Theory Ramifications, 2001, 18(5): 795-812

[20] Zhang, Y., Kauffman, L. H. and Molin, Ge, Virtual extension of Temperley-Lieb algebra, arXiv: math-ph/0610052.

[21] Zhang, Y., Kauffman, L. H. and Werner, R. F., Permutation and its partial transpose, accepted by International Journal of Quantum Information for publication, arXiv: quant-ph/0606005.