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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

Citations:

Zbl 1070.57008
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References:

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