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

Tanaka, Kokoro

doi:10.1090/S0002-9939-06-08397-3

Khovanov-Jacobsson numbers and invariants of surface-knots derived from Bar-Natan’s theory
HTML articles powered by AMS MathViewer

by Kokoro Tanaka PDF

Proc. Amer. Math. Soc. 134 (2006), 3685-3689 Request permission

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

Dror Bar-Natan, On Khovanov’s categorification of the Jones polynomial, Algebr. Geom. Topol. 2 (2002), 337–370. MR 1917056, DOI 10.2140/agt.2002.2.337
Dror Bar-Natan, Khovanov’s homology for tangles and cobordisms, Geom. Topol. 9 (2005), 1443–1499. MR 2174270, DOI 10.2140/gt.2005.9.1443
J. Scott Carter and Masahico Saito, Knotted surfaces and their diagrams, Mathematical Surveys and Monographs, vol. 55, American Mathematical Society, Providence, RI, 1998. MR 1487374, DOI 10.1090/surv/055
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).
Fujitsugu Hosokawa and Akio Kawauchi, Proposals for unknotted surfaces in four-spaces, Osaka Math. J. 16 (1979), no. 1, 233–248. MR 527028
Fujitsugu Hosokawa, T\B{o}ru Maeda, and Shin’ichi Suzuki, Numerical invariants of surfaces in $4$-space, Math. Sem. Notes Kobe Univ. 7 (1979), no. 2, 409–420. MR 557313
Magnus Jacobsson, An invariant of link cobordisms from Khovanov homology, Algebr. Geom. Topol. 4 (2004), 1211–1251. MR 2113903, DOI 10.2140/agt.2004.4.1211
Mikhail Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000), no. 3, 359–426. MR 1740682, DOI 10.1215/S0012-7094-00-10131-7
Mikhail Khovanov, An invariant of tangle cobordisms, Trans. Amer. Math. Soc. 358 (2006), no. 1, 315–327. MR 2171235, DOI 10.1090/S0002-9947-05-03665-2
M. Khovanov, Link homology and Frobenius extensions, preprint (math.QA/0411447).
Eun Soo Lee, An endomorphism of the Khovanov invariant, Adv. Math. 197 (2005), no. 2, 554–586. MR 2173845, DOI 10.1016/j.aim.2004.10.015
J. A. Rasmussen, Khovanov homology and the slice genus, preprint (math.GT/0402131).
J. A. Rasmussen, Khovanov’s invariant for closed surfaces, preprint (math.GT/0502527).
K. Tanaka, Khovanov-Jacobsson numbers and invariants of surface-knots derived from Bar-Natan’s theory, pre-publication version (math.GT/0502371).
S. M. Wehrli, Khovanov homology and Conway mutation, preprint (math.GT/0301312).