Acta Applicandae Mathematica

, Volume 83, Issue 3, pp 221–233 | Cite as

Virtual Knots and Infinite-Dimensional Lie Algebras

  • Vassily O. Manturov


In the present work, we construct an invariant of virtual knots valued in (infinite-dimensional) Lie Algebras and establish some properties of it. This leads to some heuristic ideas how to construct quandles and extract (virtual) link invariants.

knot virtual knot Lie algebra Campbell–Hausdorff formula 


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  1. 1.
    Adams, C.: The Knot Book, Freeman & Co., New York, 1994.Google Scholar
  2. 2.
    Carter, J. S., Elhamdadi, M., Graña, M. and Saito, M.: Cocycle knot invariants from quandle modules and generalized quandle homology, arXiv: math/0306068 v.1./12 Aug. 2003.Google Scholar
  3. 3.
    Dynkin, E. B.: On the coefficients in Campbell—Hausdorff formula, Dokl. Math. 57(4) (1947), 323-326.Google Scholar
  4. 4.
    Goussarov, M., Polyak, M. and Viro, O.: Finite type invariants of classical and virtual knots, Topology 39(2000), 1045-1068.Google Scholar
  5. 5.
    Joyce, D.: A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23(1) (1982), 37-65.Google Scholar
  6. 6.
    Kauffman, L. H.: Virtual knot theory, European J. Combin. 20(7) (1999), 662-690.Google Scholar
  7. 7.
    Kim, S.-G.: Virtual knot groups, arXiv:math.GT/9907172, v.1, 1999.Google Scholar
  8. 8.
    Kauffman, L. H. and Manturov, V. O.: Virtual biquandles, in preparation.Google Scholar
  9. 9.
    Manturov, V. O.: Knot Theory, CRC-Press/Chapman & Hall, 2004.Google Scholar
  10. 10.
    Manturov, V. O.: On the invariants of virtual links, Acta Appl. Math. 72(3) (2002), 295-309.Google Scholar
  11. 11.
    Manturov, V. O.: Invariants of virtual links, Dokl. Math. 384(1), 11-13.Google Scholar
  12. 12.
    Manturov, V. O.: Two-variable invariant polynomial for virtual links, Russian Math. Surveys 5(2002), 141-142.Google Scholar
  13. 13.
    Manturov, V. O.: Curves on surfaces, virtual knots, and the Jones—Kauffman polynomial, Dokl. Math., to appear.Google Scholar
  14. 14.
    Manturov, V. O.: Long virtual knots and their invariants, J. Knot Theory Ramifications, to appear.Google Scholar
  15. 15.
    S. V. Matveev, Distributivnye gruppoidy v teorii uzlov, Mat. Sb. 119(1) (1982), 78-88 (in Russian); English version: Distributive grouppoids in knot theory, Math. USSR-Sb. 47(1984), 73-83.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Vassily O. Manturov
    • 1
  1. 1.Russia. e-mail

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