Algebraic & Geometric Topology

The foam and the matrix factorization $\mathit{sl}_3$ link homologies are equivalent

Marco Mackaay and Pedro Vaz

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Abstract

We prove that the universal rational sl3 link homologies which were constructed by Khovanov in [?] and the authors in [?], using foams, and by Khovanov and Rozansky in [?], using matrix factorizations, are naturally isomorphic as projective functors from the category of links and link cobordisms to the category of bigraded vector spaces.

Article information

Source
Algebr. Geom. Topol., Volume 8, Number 1 (2008), 309-342.

Dates
Received: 16 October 2007
Revised: 5 November 2007
Accepted: 2 January 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796815

Digital Object Identifier
doi:10.2140/agt.2008.8.309

Mathematical Reviews number (MathSciNet)
MR2443231

Zentralblatt MATH identifier
1159.57005

Subjects
Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 81R50: Quantum groups and related algebraic methods [See also 16T20, 17B37] 18G60: Other (co)homology theories [See also 19D55, 46L80, 58J20, 58J22]

Keywords
$sl_3$ foams Khovanov Khovanov–Rozansky link homology matrix factorization

Citation

Mackaay, Marco; Vaz, Pedro. The foam and the matrix factorization $\mathit{sl}_3$ link homologies are equivalent. Algebr. Geom. Topol. 8 (2008), no. 1, 309--342. doi:10.2140/agt.2008.8.309. https://projecteuclid.org/euclid.agt/1513796815


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References

  • B Gornik, Note on Khovanov link cohomology
  • M-J Jeong, D Kim, Quantum $sl(n,\mathbb{C})$ link invariants
  • M Khovanov, $sl(3)$ link homology, Algebr. Geom. Topol. 4 (2004) 1045–1081
  • M Khovanov, Link homology and categorification, from: “International Congress of Mathematicians. Vol. II”, Eur. Math. Soc., Zürich (2006) 989–999
  • M Khovanov, L Rozansky, Virtual crossings, convolutions and a categorification of the $SO(2N)$ Kauffman polynomial
  • G Kuperberg, Spiders for rank 2 Lie algebras, Comm. Math. Phys. 180 (1996) 109–151
  • M Mackaay, P Vaz, The universal $\mathrm{sl}_3$ link homology, Algebr. Geom. Topol. 7 (2007) 1135–1169
  • S Morrison, A Nieh, On Khovanov's cobordism theory for $su(3)$ knot homology
  • J Rasmussen, Some differentials on Khovanov–Rozansky homology
  • H Wu, On the quantum filtration for the Khovanov–Rozansky cohomology