Abstract
Suppose \(\Sigma \) is a compact oriented surface (possibly with boundary) that has genus zero, and L is a link in the interior of \((-1,1)\times \Sigma \). We prove that the Asaeda–Przytycki–Sikora (APS) homology of L has rank 2 if and only if L is isotopic to an embedded knot in \(\{0\}\times \Sigma \). As a consequence, the APS homology detects the unknot in \((-1,1)\times \Sigma \). This is the first detection result for generalized Khovanov homology that is valid on an infinite family of manifolds, and it partially solves a conjecture in Xie and Zhang (Instantons and Khovanov skein homology on \(I\times T^2\), 2020. arXiv:2005.12863). Our proof is different from the previous detection results obtained by instanton homology because in this case, the second page of Kronheimer–Mrowka’s spectral sequence is not isomorphic to the APS homology. We also characterize all links in product manifolds that have minimal sutured instanton homology, which may be of independent interest.
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Notes
When \(\Sigma \) is non-orientable, there is a unique \((-1,1)\)–bundle over \(\Sigma \) such that the total manifold is orientable, and the APS homology can be defined for links in this bundle. We will only consider the case when \(\Sigma \) is oriented in this paper.
This is achieved by specifying a canonical reducible connection on the bundle. For details, the reader may refer to [14], see the paragraph containing Equation (12) and the discussion after Definition 3.7.
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Acknowledgements
We would like to thank Ciprian Manolescu for helpful comments, and thank Lvzhou Chen for helpful discussions.
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The second author was supported by National Key R &D Program of China 2020YFA0712800 and NSFC 12071005.
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