A rank inequality for the annular Khovanov homology of 2-periodic links. (English) Zbl 1394.57009

A link \(\tilde{L}\subset S^3\) is called \(2\)-periodic, if there is an involution \(\tau\) of \(S^3\) whose fixed set is an unknot, and which restricts to a diffeomorphism of \(\tilde{L}\). Deleting the fixed set from \(S^3\) allows us to view \(\tilde{L}\) as an annular link, that is, embedded in a thickened annulus. From this, there is a well defined procedure to produce a quotient link \(L\), which is again annular.
For annular links, M. M. Asaeda et al. [Algebr. Geom. Topol. 4, 1177–1210 (2004; Zbl 1070.57008)] defined annular Khovanov homology, a triple graded homology theory \(\mathrm{AKh}^{j,k}(L)\), where \(j\) is a quantum grading, and \(k\) an \(\mathfrak{sl}_2\) weight-space grading. This comes with a spectral sequence to ordinary Khovanov homology \(\mathrm{Kh}^j(L)\).
In this paper the author establishes a spectral sequence starting from \(\mathrm{AKh}^{2j-k,k}(\tilde{L})\otimes_{\mathbb{F}_2} \mathbb{F}_2[\theta,\theta^{-1}]\) converging to \(\mathrm{AKh}^{j,k}(L)\otimes_{\mathbb{F}_2}\mathbb{F}_2[\theta,\theta^{-1}]\) for a \(2\)-periodic link \(\tilde{L}\) with quotient link \(L\). She also conjectures that there is a spectral sequence where the annular Khovanov homology of \(\tilde{L}\) is replaced by the Khovanov homology of \(\tilde{L}\), and still converging to the annular Khovanov homology of \(L\). This would imply that \(\mathrm{rk}_{\mathbb{F}_2}\mathrm{Kh}(\tilde{L})\geq \mathrm{rk}_{\mathbb{F}_2}\mathrm{AKh}(L)\). While this may be somewhat surprising, the author gathers evidence for this to hold, including a proof of the existence of the spectral sequence for a braid closure \(L\) which has a diagram with at most one positive crossing.
The paper is carefully written, with the necessary constructions well explained and proofs given in detail.


57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M60 Group actions on manifolds and cell complexes in low dimensions


Zbl 1070.57008
Full Text: DOI arXiv


[1] 10.2140/agt.2004.4.1177 · Zbl 1070.57008
[2] 10.1090/S0002-9947-2015-06252-7 · Zbl 1365.57011
[3] 10.1093/imrn/rnq220 · Zbl 1231.57019
[4] ; Chbili, Kobe J. Math., 27, 73, (2010)
[5] 10.1112/S0010437X17007540 · Zbl 1422.57036
[6] 10.2140/agt.2012.12.2127 · Zbl 1277.53093
[7] 10.4310/JSG.2015.v13.n3.a2 · Zbl 1343.53086
[8] 10.1215/S0012-7094-00-10131-7 · Zbl 0960.57005
[9] 10.4171/JEMS/590 · Zbl 1342.57013
[10] ; McCleary, A user’s guide to spectral sequences. Cambridge Studies in Adv. Math., 58, (2001) · Zbl 0959.55001
[11] 10.1007/BF02566836 · Zbl 0206.25603
[12] 10.2140/pjm.1988.131.319 · Zbl 0661.57001
[13] 10.1142/S0218216517410073 · Zbl 1376.57011
[14] 10.1142/9789812796073_0020
[15] 10.2140/gt.2013.17.413 · Zbl 1415.57009
[16] 10.1215/S0012-7094-06-13432-4 · Zbl 1108.57011
[17] 10.1007/s00039-010-0099-y · Zbl 1210.53084
[18] 10.1016/0040-9383(93)90022-N · Zbl 0782.57006
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