Bundles of coloured posets and a Leray-Serre spectral sequence for Khovanov homology
HTML articles powered by AMS MathViewer
- by Brent Everitt and Paul Turner PDF
- Trans. Amer. Math. Soc. 364 (2012), 3137-3158 Request permission
Abstract:
The decorated hypercube found in the construction of Khovanov homology for links is an example of a Boolean lattice equipped with a presheaf of modules. One can place this in a wider setting as an example of a coloured poset, that is to say, a poset with a unique maximal element equipped with a presheaf of modules. In this paper we initiate the study of a bundle theory for coloured posets, producing for a certain class of base posets a Leray-Serre type spectral sequence. We then show how this theory finds an application in Khovanov homology by producing a new spectral sequence converging to the Khovanov homology of a given link.References
- Dror Bar-Natan, On Khovanov’s categorification of the Jones polynomial, Algebr. Geom. Topol. 2 (2002), 337–370. MR 1917056, DOI 10.2140/agt.2002.2.337
- Brent Everitt and Paul Turner, Homology of coloured posets: a generalisation of Khovanov’s cube construction, J. Algebra 322 (2009), no. 2, 429–448. MR 2529096, DOI 10.1016/j.jalgebra.2009.04.005
- James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
- Mikhail Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000), no. 3, 359–426. MR 1740682, DOI 10.1215/S0012-7094-00-10131-7
- Emmanuel Wagner, Sur l’homologie de Khovanov-Rozansky des graphes et des entrelacs, Ph.D. Thesis (2008).
Additional Information
- Brent Everitt
- Affiliation: Department of Mathematics, University of York, York YO10 5DD, England
- Email: brent.everitt@york.ac.uk
- Paul Turner
- Affiliation: Département de mathématiques, Université de Fribourg, CH-1700 Fribourg, Switzerland – and – Section de mathématiques, Université de Genève, 2-4 rue du Lièvre, CH-1211, Geneva, Switzerland
- Email: prt.maths@gmail.com
- Received by editor(s): January 16, 2009
- Received by editor(s) in revised form: August 25, 2010
- Published electronically: January 31, 2012
- Additional Notes: The first author thanks Finnur Larusson for many useful and stimulating discussions. He is also grateful to the Alpine Mathematical Institute, Haute-Savoie, France, and to the Institute for Geometry and its Applications, University of Adelaide, Australia.
The second author was partially supported by the Swiss National Science Foundation projects no. 200020-113199 and no. 200020-121506. - © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 3137-3158
- MSC (2010): Primary 57M27; Secondary 06A11, 55T10
- DOI: https://doi.org/10.1090/S0002-9947-2012-05459-6
- MathSciNet review: 2888240