2022/10/29 by David Boozer, Boozer, David
Mathematics · Medicine · #53D40 #57M27 #57R58 #Botulinum Toxin and Related Neurological Disorders #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2210.16452
openalex publication_date 2022/10/29 · openalex created_date 2022/11/06 · openalex updated_date 2026/07/28
We describe a strategy for constructing reduced Khovanov homology for links in lens spaces by generalizing a symplectic interpretation of reduced Khovanov homology for links in S3 due to Hedden, Herald, Hogancamp, and Kirk. The strategy relies on a partly conjectural description of the Fukaya category of the traceless SU(2) character variety of the 2-torus with two punctures. From a diagram of a 1-tangle in a solid torus, we construct a corresponding object (X,δ) in the A_∞ category of twisted complexes over this Fukaya category. The homotopy type of (X,δ) is an isotopy invariant of the tangle diagram. We use (X,δ) to construct cochain complexes for links in S3 and some links in S2 × S1. For links in S3, the cohomology of our cochain complex reproduces reduced Khovanov homology, though the cochain complex itself is not the usual one. For links in S2 × S1, we present results that suggest the cohomology of our cochain complex may be a link invariant.