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An oriented model for Khovanov homology. (English) Zbl 1195.57024

Khovanov homology is a categorification of the Jones polynomial, that is, its graded Euler characteristic is equal to the Jones polynomial. It was proved to be functorial up to sign with respect to link cobordism by D. Bar-Natan [Geom. Topol. 9, 1443–1499 (2005; Zbl 1084.57011)], M. Jacobsson [Algebr. Geom. Topol. 4, 1211–1251 (2004; Zbl 1072.57018) and Trans. Am. Math. Soc. 358, No. 1, 315–327 (2006; Zbl 1084.57021)]. D. Clark, S. Morrison and K. Walker [Geom. Topol. 13, No. 3, 1499–1582 (2009; Zbl 1169.57012)] and C. Caprau [Algebr. Geom. Topol. 8, No. 2, 729–756 (2008; Zbl 1148.57016)] resolved the sign ambiguity by introducing a variant of Khovanov homology with coefficients in the Gaussian integers \(\mathbb{Z}[\sqrt{-1}]\).
In the paper under review the author gives another variant of Khovanov homology with coefficients in \(\mathbb{Z}\), which is also functorial with respect to link cobordism. To do this, he introduces the trivalent category involving trivalent planar graphs and trivalent surfaces by using the graphical calculus for the HOMFLY-pt polynomial by T. Ohtsuki, S. Yamada and the reviewer [Enseign. Math., II. Ser. 44, No. 3–4, 325–360 (1998; Zbl 0958.57014)].

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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