Research ArticlesNo Access

The Bar-Natan theory splits

Yuval Wigderson

Department of Mathematics, Princeton University, USA

https://doi.org/10.1142/S0218216516500140Cited by:3 (Source: Crossref)

Abstract

We show that over the binary field $𝔽_{2}$ , the Bar-Natan perturbation of Khovanov homology splits as the direct sum of its two reduced theories, which we also prove are isomorphic. This extends Shumakovitch’s analogous result for ordinary Khovanov homology, without the perturbation.

Keywords:

AMSC: 57M25, 57M27

References

1. D. Bar-Natan, On Khovanov’s categorification of the Jones polynomial, Algebraic Geom. Topol. 2 (2002) 337–370. Crossref, Google Scholar
2. D. Bar-Natan, Khovanov’s homology for tangles and cobordisms, Geom. Topol. 9 (2005) 1443–1499. Crossref, Web of Science, Google Scholar
3. M. Jacobsson, An invariant of link cobordisms from Khovanov homology, Algebraic Geome. Topol. 4(2) (2004) 1211–1251. Crossref, Google Scholar
4. M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101(3) (2000) 359–426. Crossref, Web of Science, Google Scholar
5. M. Khovanov, A functor-valued invariant of tangles, Algebr. Geom. Topol. 2 (2002) 665–741. Crossref, Google Scholar
6. M. Khovanov, Patterns in knot cohomology I, Experi. Math. 12(3) (2003) 365–374. Crossref, Web of Science, Google Scholar
7. M. Khovanov and L. Rozansky, Matrix factorizations and link homology, Fund. Math. 199(1) (2008) 1–91. Crossref, Web of Science, Google Scholar
8. M. Khovanov and L. Rozansky, Matrix factorizations and link homology II, Geom. Topol. 12(3) (2008) 1387–1425. Crossref, Web of Science, Google Scholar
9. E. S. Lee, An endomorphism of the Khovanov invariant, Adv. Math. 197(2) (2005) 554–586. Crossref, Web of Science, Google Scholar
10. P. Ozsváth, J. Rasmussen and Z. Szabó, Odd Khovanov homology, preprint (2007), arXiv:0710.4300. Google Scholar
11. A. Shumakovitch, Torsion of the Khovanov homology, preprint (2004), arXiv:math/0405474. Google Scholar
12. Z. Szabó, A geometric spectral sequence in Khovanov homology, preprint (2010), arXiv:1010.4252. Google Scholar