Resumen de Instanton homology and knot detection on thickened surfaces

Zhenkun Li, Yi Xie, Boyu Zhang

• Suppose is a compact oriented surface (possibly with boundary) that has genus zero, and L is a link in the interior of (−1, 1) × . 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} × . As a consequence, the APS homology detects the unknot in (−1, 1) × .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 × 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.