Duke Mathematical Journal

A refinement of Rasmussen’s S-invariant

Robert Lipshitz and Sucharit Sarkar

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In a previous work, we constructed a spectrum-level refinement of Khovanov homology. This refinement induces stable cohomology operations on Khovanov homology. In this paper we show that these cohomology operations commute with cobordism maps on Khovanov homology. As a consequence we obtain a refinement of Rasmussen’s slice genus bound s for each stable cohomology operation. We show that in the case of the Steenrod square Sq2 our refinement is strictly stronger than s.

Article information

Duke Math. J., Volume 163, Number 5 (2014), 923-952.

First available in Project Euclid: 26 March 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 55P42: Stable homotopy theory, spectra


Lipshitz, Robert; Sarkar, Sucharit. A refinement of Rasmussen’s S -invariant. Duke Math. J. 163 (2014), no. 5, 923--952. doi:10.1215/00127094-2644466. https://projecteuclid.org/euclid.dmj/1395856219

Export citation


  • [1] D. Bar-Natan, Khovanov’s homology for tangles and cobordisms, Geom. Topol. 9 (2005), 1443–1499.
  • [2] R. L. Cohen, J. D. S. Jones, and G. B. Segal, Floer’s Infinite-Dimensional Morse Theory and Homotopy Theory, Progr. Math. 133, Birkhäuser, Basel, 1995, 297–325.
  • [3] M. Freedman, R. Gompf, S. Morrison, and K. Walker, Man and machine thinking about the smooth 4-dimensional Poincaré conjecture, Quantum Topol. 1 (2010), 171–208.
  • [4] M. Jacobsson, An invariant of link cobordisms from Khovanov homology, Algebr. Geom. Topol. 4 (2004), 1211–1251 (electronic).
  • [5] The Knot Atlas, http://katlas.org/Data/.
  • [6] M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000), 359–426.
  • [7] Mikhail Khovanov, A functor-valued invariant of tangles, Algebr. Geom. Topol. 2 (2002), 665–741.
  • [8] Mikhail Khovanov, An invariant of tangle cobordisms, Trans. Amer. Math. Soc. 358 (2006), 315–327.
  • [9] KnotTheory, http://katlas.org/wiki/The_Mathematica_Package_KnotTheory.
  • [10] P. B. Kronheimer and T. S. Mrowka, Gauge theory and Rasmussen’s invariant, J. Topol. 6 (2013), 659–674.
  • [11] E. S. Lee, An endomorphism of the Khovanov invariant, Adv. Math. 197 (2005), 554–586.
  • [12] R. Lipshitz and S. Sarkar, A Khovanov stable homotopy type, to appear in J. Amer. Math. Soc., preprint, arXiv:1112.3932v4 [math.GT].
  • [13] Robert Lipshitz and Sucharit Sarkar, A Steenrod square on Khovanov homology, to appear in J. Topol., preprint, arXiv:1204.5776v3 [math.GT].
  • [14] M. Mackaay, P. Turner, and P. Vaz, A remark on Rasmussen’s invariant of knots, J. Knot Theory Ramifications 16 (2007), 333–344.
  • [15] J. Rasmussen, Khovanov homology and the slice genus, Invent. Math. 182 (2010), 419–447.
  • [16] C. Seed, Knotkit, https://github.com/cseed/knotkit.
  • [17] P. R. Turner, Calculating Bar-Natan’s characteristic two Khovanov homology, J. Knot Theory Ramifications 15 (2006), 1335–1356.