## The Michigan Mathematical Journal

### Equivariant Khovanov Homology of Periodic Links

Wojciech Politarczyk

#### Abstract

The purpose of this paper is to construct and study equivariant Khovanov homology, a version of Khovanov homology theory for periodic links. Since our construction works regardless of the characteristic of the coefficient ring, it generalizes a previous construction by Chbili. We establish invariance under equivariant isotopies of links and study algebraic properties of integral and rational version of the homology theory. Moreover, we construct a skein spectral sequence converging to equivariant Khovanov homology and use this spectral sequence to compute, as an example, equivariant Khovanov homology of torus links $T(n,2)$.

#### Article information

Source
Michigan Math. J., Advance publication (2019), 31 pages.

Dates
Revised: 31 July 2018
First available in Project Euclid: 8 August 2019

https://projecteuclid.org/euclid.mmj/1565251218

Digital Object Identifier
doi:10.1307/mmj/1565251218

#### Citation

Politarczyk, Wojciech. Equivariant Khovanov Homology of Periodic Links. Michigan Math. J., advance publication, 8 August 2019. doi:10.1307/mmj/1565251218. https://projecteuclid.org/euclid.mmj/1565251218

#### References

• [BN05] D. Bar-Natan, Khovanov’s homology for tangles and cobordisms, Geom. Topol. 9 (2005), 1443–1499.
• [BP18] M. Borodzik and W. Politarczyk, Khovanov homology and periodic links, Indiana Univ. Math. J. (2018, to appear), http://arxiv.org/abs/1704.07316, arXiv:1704.07316.
• [Chb10] N. Chbili, Equivalent Khovanov homology associated with symmetric links, Kobe J. Math. 27 (2010), no. 1–2, 73–89.
• [Cor16] J. Cornish, Sutured annular Khovanov homology and two periodic braids, 2016, http://arxiv.org/abs/1606.03034, arXiv:1606.03034.
• [CR90] C. W. Curtis and I. Reiner, Methods of representation theory. Vol. I, Wiley Classics Library, Wiley, New York, 1990, With applications to finite groups and orders, Reprint of the 1981 original, A Wiley-Interscience Publication.
• [ET12] B. Everitt and P. Turner, Bundles of coloured posets and a Leray–Serre spectral sequence for Khovanov homology, Trans. Amer. Math. Soc. 364 (2012), no. 6, 3137–3158.
• [Hen15] K. Hendricks, Localization of the link Floer homology of doubly-periodic knots, J. Symplectic Geom. 13 (2015), no. 3, 545–608.
• [HLS16] K. Hendricks, R. Lipshitz, and S. Sarkar, A flexible construction of equivariant Floer homology and applications, J. Topol. 9 (2016), no. 4, 1153–1236.
• [IR90] K. Ireland and M. Rosen, A classical introduction to modern number theory, Second edition, Grad. Texts in Math., 84, Springer-Verlag, New York, 1990.
• [Kho00] M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000), no. 3, 359–426.
• [Lee05] E. S. Lee, An endomorphism of the Khovanov invariant, Adv. Math. 197 (2005), no. 2, 554–586.
• [McC01] J. McCleary, A user’s guide to spectral sequences, Second edition, Cambridge Stud. Adv. Math., 58, Cambridge University Press, Cambridge, 2001.
• [MB84] J. W. Morgan and H. Bass eds., The Smith conjecture, Pure Appl. Math., 112, Academic Press, Inc., Orlando, FL, 1984, Papers presented at the symposium held at Columbia University, New York, 1979.
• [Mur71] K. Murasugi, On periodic knots, Comment. Math. Helv. 46 (1971), 162–174.
• [Mur88] K. Murasugi, Jones polynomials of periodic links, Pacific J. Math. 131 (1988), no. 2, 319–329.
• [Pol17] W. Politarczyk, Equivariant Jones polynomials of periodic links, J. Knot Theory Ramifications 26 (2017), no. 03, 1741007.
• [Pol15] W. Politarczyk, Khovanov homology of symmetric links, Ph.D. Thesis, Adam Mickiewicz University in Poznań, 2015, https://repozytorium.amu.edu.pl/bitstream/10593/13034/1/politarczyk_phd.pdf.
• [Prz89] J. H. Przytycki, On Murasugi’s and Traczyk’s criteria for periodic links, Math. Ann. 283 (1989), no. 3, 465–478.
• [Ras10] J. Rasmussen, Khovanov homology and the slice genus, Invent. Math. 182 (2010), no. 2, 419–447.
• [SS10] P. Seidel and I. Smith, Localization for involutions in Floer cohomology, Geom. Funct. Anal. 20 (2010), no. 6, 1464–1501.
• [TSPA14] The Stacks Project Authors, Stacks Project – Derived Categories, 2014, http://stacks.math.columbia.edu/download/derived.pdf.
• [tD87] T. tom Dieck, Transformation groups, de Gruyter Stud. Math., 8, Walter de Gruyter & Co., Berlin, 1987.
• [Tra90a] P. Traczyk, $10_{101}$ has no period $7$: a criterion for periodic links, Proc. Amer. Math. Soc. 108 (1990a), no. 3, 845–846.
• [Tra90b] P. Traczyk, A criterion for knots of period $3$, Topology Appl. 36 (1990b), no. 3, 275–281.
• [Tra91] P. Traczyk, Periodic knots and the skein polynomial, Invent. Math. 106 (1991), no. 1, 73–84.
• [Tur08] P. Turner, A spectral sequence for Khovanov homology with an application to $(3,q)$-torus links, Algebr. Geom. Topol. 8 (2008), no. 2, 869–884.
• [Tur17] P. Turner, Five lectures on Khovanov homology, J. Knot Theory Ramifications 26 (2017), no. 03.
• [Vog15] P. Vogel, Functoriality of Khovanov homology, 2015, arXiv:1505.04545.
• [Wal79] C. T. C. Wall, Periodic projective resolutions, Proc. Lond. Math. Soc. (3) 39 (1979), no. 3, 509–553.
• [Wei94] C. A. Weibel, An introduction to homological algebra, Cambridge Stud. Adv. Math., 38, Cambridge University Press, Cambridge, 1994.
• [Zha18] M. Zhang, A rank inequality for the annular Khovanov homology of 2-periodic links, Algebr. Geom. Topol. 18 (2018), no. 2, 1147–1194.