Algebraic & Geometric Topology

The link concordance invariant from Lee homology

John Pardon

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We use the knot homology of Khovanov and Lee to construct link concordance invariants generalizing the Rasmussen s–invariant of knots. The relevant invariant for a link is a filtration on a vector space of dimension 2|L|. The basic properties of the s–invariant all extend to the case of links; in particular, any orientable cobordism Σ between links induces a map between their corresponding vector spaces which is filtered of degree χ(Σ). A corollary of this construction is that any component-preserving orientable cobordism from a Kh–thin link to a link split into k components must have genus at least k2. In particular, no quasi-alternating link is concordant to a split link.

Article information

Algebr. Geom. Topol., Volume 12, Number 2 (2012), 1081-1098.

Received: 25 July 2011
Revised: 9 February 2012
Accepted: 14 February 2012
First available in Project Euclid: 19 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M27: Invariants of knots and 3-manifolds 57Q60: Cobordism and concordance

Khovanov homology link concordance link cobordism Rasmussen s-invariant slice genus


Pardon, John. The link concordance invariant from Lee homology. Algebr. Geom. Topol. 12 (2012), no. 2, 1081--1098. doi:10.2140/agt.2012.12.1081.

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  • D Bar-Natan, On Khovanov's categorification of the Jones polynomial, Algebr. Geom. Topol. 2 (2002) 337–370
  • D Bar-Natan, Khovanov's homology for tangles and cobordisms, Geom. Topol. 9 (2005) 1443–1499
  • D Bar-Natan, Fast Khovanov homology computations, J. Knot Theory Ramifications 16 (2007) 243–255
  • D Bar-Natan, S Morrison, KnotTheory\` Package (2008) Available at \setbox0\makeatletter\@url {\unhbox0
  • A Beliakova, S Wehrli, Categorification of the colored Jones polynomial and Rasmussen invariant of links, Canad. J. Math. 60 (2008) 1240–1266
  • C L Caprau, $\rm sl(2)$ tangle homology with a parameter and singular cobordisms, Algebr. Geom. Topol. 8 (2008) 729–756
  • D Clark, S Morrison, K Walker, Fixing the functoriality of Khovanov homology, Geom. Topol. 13 (2009) 1499–1582
  • M Freedman, R Gompf, S Morrison, K Walker, Man and machine thinking about the smooth 4-dimensional Poincaré conjecture, Quantum Topol. 1 (2010) 171–208
  • S Friedl, M Powell, Cobordisms to weakly splittable links
  • J Greene, Homologically thin, non-quasi-alternating links, Math. Res. Lett. 17 (2010) 39–49
  • A E Hatcher, A proof of the Smale conjecture, ${\rm Diff}(S^{3})\simeq {\rm O}(4)$, Ann. of Math. (2) 117 (1983) 553–607
  • M Jacobsson, An invariant of link cobordisms from Khovanov homology, Algebr. Geom. Topol. 4 (2004) 1211–1251
  • A Kawauchi, On alternation numbers of links, Topology Appl. 157 (2010) 274–279
  • M Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359–426
  • P B Kronheimer, T S Mrowka, Gauge theory for embedded surfaces, I, Topology 32 (1993) 773–826
  • P B Kronheimer, T S Mrowka, The genus of embedded surfaces in the projective plane, Math. Res. Lett. 1 (1994) 797–808
  • P B Kronheimer, T S Mrowka, Gauge theory for embedded surfaces, II, Topology 34 (1995) 37–97
  • E S Lee, An endomorphism of the Khovanov invariant, Adv. Math. 197 (2005) 554–586
  • C Livingston, Computations of the Ozsváth–Szabó knot concordance invariant, Geom. Topol. 8 (2004) 735–742
  • C Manolescu, P Ozsváth, On the Khovanov and knot Floer homologies of quasi-alternating links, from: “Proceedings of Gökova Geometry-Topology Conference 2007”, (S Akbulut, T Önder, R J Stern, editors), Gökova Geometry/Topology Conference (2008) 60–81
  • K Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117 (1965) 387–422
  • P Ozsváth, Z Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004) 58–116
  • P Ozsváth, Z Szabó, On the Heegaard Floer homology of branched double-covers, Adv. Math. 194 (2005) 1–33
  • P Ozsváth, Z Szabó, Holomorphic disks, link invariants and the multi-variable Alexander polynomial, Algebr. Geom. Topol. 8 (2008) 615–692
  • J Rasmussen, Khovanov's invariant for closed surfaces
  • J Rasmussen, Khovanov homology and the slice genus, Invent. Math. 182 (2010) 419–447
  • K Walker (mathoverflownet/users/284), Is the complete functorial structure for Khovanov–Lee homology known? (accessed 2011-07-22) Available at \setbox0\makeatletter\@url {\unhbox0