|
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
Posted:
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  -knot has trivial Khovanov-Jacobsson number.
References
- 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
 -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:
http://dx.doi.org/10.1090/S0002-9939-06-08397-3
PII:
S 0002-9939(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
Posted:
May 18, 2006
Communicated by:
Ronald A. Fintushel
Article copyright:
© Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain after
28 years from publication.
|
