## 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.

### MSC:

 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
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