2022/08/30 by Zhenkun Li, Yi Xie, Li, Zhenkun +3
Computer Science · Mathematics · #57M27 #57R58 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2208.13963
openalex publication_date 2022/08/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Suppose Σ is a compact oriented surface (possibly with boundary) that has genus zero, and L is a link in the interior of (-1,1)×Σ. 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\×Σ. As a consequence, the APS homology detects the unknot in (-1,1)×Σ. 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 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.