## Algebraic & Geometric Topology

### Odd Khovanov homology

#### Abstract

We describe an invariant of links in $S 3$ which is closely related to Khovanov’s Jones polynomial homology. Our construction replaces the symmetric algebra appearing in Khovanov’s definition with an exterior algebra. The two invariants have the same reduction modulo $2$, but differ over $ℚ$. There is a reduced version which is a link invariant whose graded Euler characteristic is the normalized Jones polynomial.

#### Article information

Source
Algebr. Geom. Topol., Volume 13, Number 3 (2013), 1465-1488.

Dates
Revised: 11 July 2012
Accepted: 18 June 2012
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715589

Digital Object Identifier
doi:10.2140/agt.2013.13.1465

Mathematical Reviews number (MathSciNet)
MR3071132

Zentralblatt MATH identifier
1297.57032

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57R58: Floer homology

Keywords

#### Citation

Ozsváth, Peter S; Rasmussen, Jacob; Szabó, Zoltán. Odd Khovanov homology. Algebr. Geom. Topol. 13 (2013), no. 3, 1465--1488. doi:10.2140/agt.2013.13.1465. https://projecteuclid.org/euclid.agt/1513715589

#### References

• D Bar-Natan, On Khovanov's categorification of the Jones polynomial, Algebr. Geom. Topol. 2 (2002) 337–370
• J M Bloom, Odd Khovanov homology is mutation invariant, Math. Res. Lett. 17 (2010) 1–10
• G Burde, H Zieschang, Knots, 2nd edition, de Gruyter Studies in Math. 5, Walter de Gruyter & Co., Berlin (2003)
• M Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359–426
• M Khovanov, Patterns in knot cohomology. I, Experiment. Math. 12 (2003) 365–374
• M Khovanov, L Rozansky, Matrix factorizations and link homology, Fund. Math. 199 (2008) 1–91
• M Khovanov, L Rozansky, Matrix factorizations and link homology. II, Geom. Topol. 12 (2008) 1387–1425
• P Kronheimer, T Mrowka, Monopoles and three-manifolds, New Mathematical Monographs 10, Cambridge Univ. Press (2007)
• E S Lee, An endomorphism of the Khovanov invariant, Adv. Math. 197 (2005) 554–586
• C Manolescu, P Ozsváth, On the Khovanov and knot Floer homologies of quasi-alternating links, from: “Proc. Gökova Geometry-Topology Conf. 2007”, (S Akbulut, T Önder, R J Stern, editors), GGT, Gökova (2008) 60–81
• V O Manturov, Khovanov homology for virtual knots with arbitrary coefficients, J. Knot Theory Ramifications 16 (2007) 345–377
• P Ozsváth, Z Szabó, On the Heegaard Floer homology of branched double-covers, Adv. Math. 194 (2005) 1–33
• K Putyra, Cobordisms with chronologies and a generalisation of the Khovanov complex
• J Rasmussen, Knot polynomials and knot homologies, from: “Geometry and topology of manifolds”, (H U Boden, I Hambleton, A J Nicas, B D Park, editors), Fields Inst. Commun. 47, Amer. Math. Soc. (2005) 261–280
• A N Shumakovitch, KhoHo, computer program (2003) Available at \setbox0\makeatletter\@url https://github.com/AShumakovitch/KhoHo {\unhbox0
• A N Shumakovitch, Patterns in odd Khovanov homology, J. Knot Theory Ramifications 20 (2011) 203–222
• Z Szabó, A geometric spectral sequence on odd Khovanov homology, preprint
• P R Turner, Calculating Bar-Natan's characteristic two Khovanov homology, J. Knot Theory Ramifications 15 (2006) 1335–1356