Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Khovanov-Jacobsson numbers and invariants of surface-knots derived from Bar-Natan's theory


Author: Kokoro Tanaka
Journal: Proc. Amer. Math. Soc. 134 (2006), 3685-3689
MSC (2000): Primary 57Q45; Secondary 57M25
DOI: https://doi.org/10.1090/S0002-9939-06-08397-3
Published electronically: May 18, 2006
MathSciNet review: 2240683
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Khovanov introduced a cohomology theory for oriented classical links whose graded Euler characteristic is the Jones polynomial. Since Khovanov's theory is functorial for link cobordisms between classical links, we obtain an invariant of a surface-knot, called the Khovanov-Jacobsson number, by considering the surface-knot as a link cobordism between empty links. In this paper, we study an extension of the Khovanov-Jacobsson number derived from Bar-Natan's theory, and prove that any $ T^2$-knot has trivial Khovanov-Jacobsson number.


References [Enhancements On Off] (What's this?)

  • 1. D. Bar-Natan, On Khovanov's categorification of the Jones polynomial, Algebr. Geom. Topol. 2 (2002), 337-370. MR 1917056 (2003h:57014)
  • 2. D. Bar-Natan, Khovanov's homology for tangles and cobordisms, Geom. Topol. 9 (2005), 1443-1499.MR 2174270
  • 3. J. S. Carter and M. Saito, ``Knotted surfaces and their diagrams'', Math. Surveys and Monographs 55, Amer. Math. Soc., 1998. MR 1487374 (98m:57027)
  • 4. J. S. Carter, M. Saito and S. Satoh, Ribbon-moves for 2-knots with 1-handles attached and Khovanov-Jacobsson numbers, to appear in Proc. Amer. Math. Soc. (math.GT/0407493).
  • 5. F. Hosokawa and A. Kawauchi, Proposal for unknotted surfaces in four-space, Osaka J. Math. 16 (1979), 233-248. MR 0527028 (81c:57018)
  • 6. F. Hosokawa, T. Maeda and S. Suzuki, Numerical invariants of surfaces in $ 4$-space, Math. Sem. Notes Kobe Univ. 7 (1979), no. 2, 409-420. MR 0557313 (81b:57019)
  • 7. M. Jacobsson, An invariant of link cobordisms from Khovanov's homology theory, Algebr. Geom. Topol. 4 (2004), 1211-1251. MR 2113903 (2005k:57047)
  • 8. M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000), no. 3, 359-426. MR 1740682 (2002j:57025)
  • 9. M. Khovanov, An invariant of tangle cobordisms, Trans. Amer. Math. Soc. 358 (2006), 315-327. MR 2171235
  • 10. M. Khovanov, Link homology and Frobenius extensions, preprint (math.QA/0411447).
  • 11. E. S. Lee, An endomorphism of the Khovanov invariant, Adv. Math. 197 (2005), 554-586. MR 2173845
  • 12. J. A. Rasmussen, Khovanov homology and the slice genus, preprint (math.GT/0402131).
  • 13. J. A. Rasmussen, Khovanov's invariant for closed surfaces, preprint (math.GT/0502527).
  • 14. K. Tanaka, Khovanov-Jacobsson numbers and invariants of surface-knots derived from Bar-Natan's theory, pre-publication version (math.GT/0502371).
  • 15. S. M. Wehrli, Khovanov homology and Conway mutation, preprint (math.GT/0301312).

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 57Q45, 57M25

Retrieve articles in all journals with MSC (2000): 57Q45, 57M25


Additional Information

Kokoro Tanaka
Affiliation: Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro, Tokyo 153-8914, Japan
Email: k-tanaka@ms.u-tokyo.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-06-08397-3
Keywords: Khovanov cohomology, surface-knot, Khovanov-Jacobsson number
Received by editor(s): March 14, 2005
Received by editor(s) in revised form: June 14, 2005
Published electronically: May 18, 2006
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society