Geometry & Topology

$\mathfrak{sl}(N)$–link homology ($N\geq 4$) using foams and the Kapustin–Li formula

Marco Mackaay, Marko Stošić, and Pedro Vaz

Full-text: Open access


We use foams to give a topological construction of a rational link homology categorifying the sl(N) link invariant, for N4. To evaluate closed foams we use the Kapustin–Li formula adapted to foams by Khovanov and Rozansky [Adv. Theor. Math. Phys. 11 (2007) 233-259]. We show that for any link our homology is isomorphic to the Khovanov–Rozansky [Fund. Math. 199 (2008) 1-91] homology.

Article information

Geom. Topol., Volume 13, Number 2 (2009), 1075-1128.

Received: 4 December 2007
Revised: 1 August 2008
Accepted: 31 December 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

\mathfraksl(N) foams link homology Kapustin–Li Khovanov–Rozansky


Mackaay, Marco; Stošić, Marko; Vaz, Pedro. $\mathfrak{sl}(N)$–link homology ($N\geq 4$) using foams and the Kapustin–Li formula. Geom. Topol. 13 (2009), no. 2, 1075--1128. doi:10.2140/gt.2009.13.1075.

Export citation


  • D Bar-Natan, Khovanov's homology for tangles and cobordisms, Geom. Topol. 9 (2005) 1443–1499
  • D Clark, S Morrison, K Walker, Fixing the functoriality of Khovanov homology
  • W Fulton, Young tableaux. With applications to representation theory and geometry, London Math. Soc. Student Texts 35, Cambridge Univ. Press (1997)
  • W Fulton, J Harris, Representation theory. A first course, Graduate Texts in Math. 129, Springer, New York (1991)
  • H Hiller, Geometry of Coxeter groups, Research Notes in Math. 54, Pitman (Advanced Publishing Program), Boston (1982)
  • A Kapustin, Y Li, Topological correlators in Landau–Ginzburg models with boundaries, Adv. Theor. Math. Phys. 7 (2003) 727–749
  • M Khovanov, sl(3) link homology, Algebr. Geom. Topol. 4 (2004) 1045–1081
  • M Khovanov, Link homology and Frobenius extensions, Fund. Math. 190 (2006) 179–190
  • M Khovanov, L Rozansky, Topological Landau–Ginzburg models on the world-sheet foam, Adv. Theor. Math. Phys. 11 (2007) 233–259
  • M Khovanov, L Rozansky, Virtual crossing, convolutions and a categorification of the ${\rm SO}(2N)$ Kauffman polynomial, J. Gökova Geom. Topol. GGT 1 (2007) 116–214
  • M Khovanov, L Rozansky, Matrix factorizations and link homology, Fund. Math. 199 (2008) 1–91
  • M Mackaay, P Vaz, The universal ${\rm sl}\sb 3$–link homology, Algebr. Geom. Topol. 7 (2007) 1135–1169
  • S Morrison, A diagrammatic category for the representation theory of $U_q(sl(N))$, PhD thesis, UC Berkeley (2007)
  • H Murakami, T Ohtsuki, S Yamada, Homfly polynomial via an invariant of colored plane graphs, Enseign. Math. $(2)$ 44 (1998) 325–360
  • L Rozansky, Topological A–models on seamed Riemann surfaces, Adv. Theor. Math. Phys. 11 (2007) 517–529