Abstract
We recall the notion of partial presimplicial set and its geometric realization. We show that any semiadequate diagram yields a partial presimplicial set leading to a geometric realization of the almost-extreme Khovanov homology of the diagram. We give a concrete formula for the homotopy type of this geometric realization, involving wedge of spheres and a suspension of the projective plane.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Categories of partial sets and pointed sets are equivalent (using natural isomorphism), see [2, Proposition 7.28].
A presimplicial module leads to the chain complex \((C_n,\partial _n)\) by defining \(\partial _n=\sum _{i=0}^n(-1)^id_i\) and further to a homology (and cohomology) module.
The geometric realization of partial presimplicial sets can be defined in various different ways; we describe here the pointed geometric realization, which we find suitable for the goal of this paper.
In the oriented version of Khovanov homology the values of J “jump” by two, and therefore \(J_{\mathrm{{almax}}}(\mathbf {D}) = J_{\max }(\mathbf {D})-2\).
We have adjusted the statement so it agrees with the framed version of Khovanov homology in Sect. 3.
Note that K is homotopy equivalent to a CW-complex with a single vertex and cells in demension n and \(n-1\), thus a Moore space.
Reader familiar with Pachner moves will notice the similarity with Pachner simplex addition [15].
References
Asaeda, M.M., Przytycki, J.H.: Khovanov homology: torsion and thickness. In: Bryden, J. (ed.) Proceedings of the Workshop, “New Techniques in Topological Quantum Field Theory” Calgary/Kananaskis, Canada, 2001 (2004). e-print: http://front.math.ucdavis.edu/math.GT/0402402
Awodey, S.: Category Theory, Oxford Logic Guides-52. Oxford University Press 2005, 2010, 2011
Cantero, F., Silvero, M.: Extreme Khovanov spectra. Rev. Mat. Iberoam. (to appear). e-print: arXiv:1803.06257 [math.GT]
Chmutov, S.: Extreme parts of the Khovanov complex, Abstract of the talk delivered at the conference Knots in Washington XXI: Skein modules, Khovanov homology and Hochschild homology, George Washington University, December 9–11, 2005 (notes to the talk are available at https://people.math.osu.edu/chmutov.1/talks/2005/Wash-XXI-2005.pdf)
Eilenberg, S., Zilber, J.: Semi-simplicial complexes and singular homology. Ann. Math. 51(2), 499–513 (1950)
Everitt, B., Turner, P.: The homotopy theory of Khovanov homology. Algbr. Geom. Topol. 14(5), 2747–2781 (2014). arXiv:1112.3460 [math.GT]
González-Meneses, J., Manchón, P.M.G., Silvero, M.: A geometric description of the extreme Khovanov cohomology. Proc. R. Soc. Edinb. Sect. A 148(3), 541–557 (2018). arXiv:1511.05845 [math.GT]
Khovanov, M.: A categorification of the Jones polynomial. Duke Math. J. 101(3), 359–426 (2000)
Knot Atlas: http://katlas.math.toronto.edu/wiki/Main_Page, maintained by D. Bar-Natan and S. Morrison. Accessed 29 June 2018
Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)
Lickorish, W.B.R., Thistlethwaite, M.B.: Some links with nontrivial polynomials and their crossing-numbers. Comment. Math. Helv. 63(1), 527–539 (1988)
Lipshitz, R., Sarkar, S.: A Khovanov stable homotopy type. J. Am. Math. Soc. 27, 983–1042 (2014)
Loday, J.L.: Cyclic homology, Grund. Math. Wissen. Band 301, Springer, Berlin, 1992 (second edition, 1998)
Pabiniak, M.D., Przytycki, J.H., Sazdanovic, R.: On the first group of the chromatic cohomology of graphs. Geom. Dedic. 140(1), 19–48 (2009). arXiv:math/0607326 [math.GT]
Pachner, U.: P.L. homeomorphic manifolds are equivalent by elementary shellings. Eur. J. Comb. 12(2), 129–145 (1991)
Przytycki, J.H.: Knots and distributive homology: from arc colorings to Yang-Baxter homology. In: New Ideas in Low Dimensional Topology. World Scientific, 56, 2015, 413-488. e-print: arXiv:1409.7044 [math.GT]
Przytycki, J.H., Sazdanovic, R.: Torsion in Khovanov homology of semi-adequate links. Fund. Math. 225, 277–303 (2014). arXiv:1210.5254 [math.QA]
Przytycki, J.H., Silvero, M.: Homotopy type of circle graphs complexes motivated by extreme Khovanov homology. J. Algebraic Comb. 48, 119–156 (2018). arXiv:1608.03002 [math.GT]
Rolfsen, D.: Knots and links, Publish or Perish, 1976 (second edition, 1990; third edition, AMS Chelsea Publishing 346, 2003)
Viro, O.: Remarks on the definition of Khovanov Homology, e-print: arXiv:math/0202199 [math.GT]
Viro, O.: Khovanov homology, its definitions and ramifications. Fund. Math. 184, 317–342 (2004)
Acknowledgements
Józef H. Przytycki is partially supported by the Simons Foundation Collaboration Grant for Mathematicians—316446 and CCAS Dean’s Research Chair award. Marithania Silvero is partially supported by Spanish Government research Projects MTM2016-76453-C2-1-P and MTM2017-86802-P, by ERC Grant PCG-336983 and by Basque Government Grant IT974-16. The authors are grateful to the Department of Mathematics of the University of Barcelona for its hospitality.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Przytycki, J.H., Silvero, M. Geometric realization of the almost-extreme Khovanov homology of semiadequate links. Geom Dedicata 204, 387–401 (2020). https://doi.org/10.1007/s10711-019-00462-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-019-00462-0