Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-19T07:05:16.657Z Has data issue: false hasContentIssue false
  • English
  • Français

Torsion in thin regions of Khovanov homology

Published online by Cambridge University Press:  29 January 2021

Alex Chandler ,
Adam M. Lowrance ,
Radmila Sazdanović  and
Victor Summers
Alex Chandler*
Affiliation:
University of Vienna, Faculty of Mathematics, Vienna, Austria
Adam M. Lowrance
Affiliation:
Vassar College, Mathematics and Statistics Department, Poughkeepsie, NY, USA e-mail: adlowrance@vassar.edu
Radmila Sazdanović
Affiliation:
North Carolina State University, Department of Mathematics, Raleigh, NC, USA e-mail: rsazdanovic@math.ncsu.edu
Victor Summers
Affiliation:
University of South Carolina Upstate, Department of Mathematics, Spartanburg, SC, USA e-mail: vsummers@uscupstate.edu
*
e-mail: alex.chandler@univie.ac.at
Get access Rights & Permissions [Opens in a new window]

Abstract

In the integral Khovanov homology of links, the presence of odd torsion is rare. Homologically thin links, that is, links whose Khovanov homology is supported on two adjacent diagonals, are known to contain only $\mathbb {Z}_2$ torsion. In this paper, we prove a local version of this result. If the Khovanov homology of a link is supported on two adjacent diagonals over a range of homological gradings and the Khovanov homology satisfies some other mild restrictions, then the Khovanov homology of that link has only $\mathbb {Z}_2$ torsion over that range of homological gradings. These conditions are then shown to be met by an infinite family of three-braids, strictly containing all three-strand torus links, thus giving a partial answer to Sazdanović and Przytycki’s conjecture that three-braids have only $\mathbb {Z}_2$ torsion in Khovanov homology. We use these computations and our main theorem to obtain the integral Khovanov homology for all links in this family.

Keywords

KnotKhovanov homologyhomologically thin
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 3 , June 2022 , pp. 630 - 654
Check for updates
Copyright
© Canadian Mathematical Society 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

AC was supported in part by the project P31705 of the Austrian Science Fund. AL was supported in part by NSF grant DMS-1811344. RS was supported in part by the Simons Foundation Collaboration Grant 318086 and NSF Grant DMS 1854705.

References

Abe, T. and Kishimoto, K., The dealternating number and the alternation number of a closed 3-braid . J. Knot Theory Ramif. 19(2010), no. 9, 1157–1181.CrossRefGoogle Scholar
Baldwin, J. A., Heegaard Floer homology and genus one, one-boundary component open books . J. Topol. 1(2008), no. 4, 963–992.CrossRefGoogle Scholar
Bar-Natan, D., Khovanov's homology for tangles and cobordisms . Geom. Topol. 9(2005), 1443–1499.CrossRefGoogle Scholar
Beliakova, A. and Wehrli, S., Categorification of the colored Jones polynomial and Rasmussen invariant of links . Can. J. Math. 60(2008), no. 6, 1240–1266.CrossRefGoogle Scholar
Benheddi, M., Khovanov homology of torus links: structure and computations. PhD thesis, University of Geneva, 2017.Google Scholar
Hatcher, A., Algebraic topology. Cambridge University Press, Cambridge, MA, 2002.Google Scholar
Helme-Guizon, L., Przytycki, J. H., and Rong, Y., Torsion in graph homology . Fundam. Math. 190(2006), 139–177.CrossRefGoogle Scholar
Jones, V. F. R., A polynomial invariant for knots via von Neumann algebras . Bull. Amer. Math. Soc., 12(1985), no. 1, 103–111.CrossRefGoogle Scholar
Khovanov, M., A categorification of the Jones polynomial . Duke Math. J. 101(2000), no. 3, 359–426.CrossRefGoogle Scholar
Lee, E. S., An endomorphism of the Khovanov invariant . Adv. Math. 197(2005), no. 2, 554–586.CrossRefGoogle Scholar
Lee, E. S., On Khovanov invariant for alternating links, Preprint, 2008. arXiv:math.GT/0210213.Google Scholar
Lipshitz, R. and Sarkar, S., A refinement of Rasmussen's $S$ -invariant . Duke Math. J. 163(2014), no. 5, 923–952.CrossRefGoogle Scholar
Lowrance, A., The Khovanov width of twisted links and closed 3-braids . Comment. Math. Helv., 86(2011), no. 3, 675–706.CrossRefGoogle Scholar
Lowrance, A. M. and Sazdanović, R., Chromatic homology, Khovanov homology, and torsion . Topol. Appl. 222(2017), 77–99.CrossRefGoogle Scholar
Manolescu, C. and Ozsváth, P., On the Khovanov and knot Floer homologies of quasi-alternating links. In: Proceedings of Gökova Geometry-Topology Conference 2007, Gökova Geometry/Topology Conference (GGT), Gökova, 2008, pp. 60–81.Google Scholar
McCleary, J., A user's guide to spectral sequences, Cambridge Studies in Advanced Mathematics, 58. 2nd ed., Cambridge University Press, Cambridge, MA, 2001.Google Scholar
Mukherjee, S., On odd torsion in even Khovanov homology . Exp. Math. (2020), 1–7.CrossRefGoogle Scholar
Mukherjee, S., Przytycki, J. H., Silvero, M., Wang, X., and Yang, S. Y., Search for torsion in Khovanov homology . Exp. Math. 27(2017), 1–10.Google Scholar
Mukherjee, S. and Schütz, D., Arbitrarily large torsion in Khovanov cohomology. To appear in Quantum Topology. Preprint, 2019. arXiv:1909.07269.Google Scholar
Murasugi, K., On closed 3-braids. Vol. 151. Memoirs of the American Mathematical Society, American Mathematical Society, Providence, RI, 1974.CrossRefGoogle Scholar
Przytycki, J. H. and Sazdanović, R., Torsion in Khovanov homology of semi-adequate links . Fundam. Math. 225(2014), no. 1, 277–304.CrossRefGoogle Scholar
Rasmussen, J., Khovanov homology and the slice genus . Invent. Math. 182(2010), no. 2, 419–447.CrossRefGoogle Scholar
Schütz, D., Torsion calculations in Khovanov cohomology . J. Knot Theory Ramif. 29(2020), no. 8, 2071001, 7.CrossRefGoogle Scholar
Shumakovitch, A. N., Torsion of Khovanov homology . Fundam. Math. 225(2014), no. 1, 343–364.CrossRefGoogle Scholar
Shumakovitch, A. N., Torsion in Khovanov homology of homologically thin knots. To appear in J. Knot Theory Ramif. Preprint, 2018. arXiv:1806.05168.Google Scholar
Stošić, M., Khovanov homology of torus links . Topol. Appl. 156(2009), no. 3, 533–541.CrossRefGoogle Scholar
Turner, P., A spectral sequence for Khovanov homology with an application to $\left(3,q\right)$ -torus links . Algebr. Geom. Topol. 8(2008), no. 2, 869–884.CrossRefGoogle Scholar
Turner, P. R., Calculating Bar-Natan's characteristic two Khovanov homology . J. Knot Theory Ramif. 15(2006), no. 10, 1335–1356.CrossRefGoogle Scholar
Viro, O., Khovanov homology, its definitions and ramifications . Fundam. Math. 184(2004), no. 317–342.CrossRefGoogle Scholar