2020/04/30 by Jianfeng Lin, Daniel Ruberman, Nikolai Saveliev
Mathematics · Medicine · #Alexander polynomial #Algebra over a field #Botulinum Toxin and Related Neurological Disorders #Combinatorics #Covering space #Floer homology #Geometric and Algebraic Topology #Geometry #Homology (biology) #Homotopy and Cohomology in Algebraic Topology #Knot (papermaking) #Knot theory #Magnetic monopole #Mathematics #Physics #Pure mathematics #Relative homology #Torus #math.DG #math.GT #msc:57M25 #msc:57R57 #msc:57R58
paper · pdf · doi:10.2140/gt.2021.25.3591
published as Geom. Topol. 25 (2021) 3591-3628 · 39 page, 2 figures. Added a reference to Langte Ma's paper arXiv:1909.01533, which contains an independent proof of our Theorem B. Final version, to appear in Geometry and Topology
arxiv created 2020/12/24 · openalex publication_date 2021/12/31 · arxiv updated 2022/02/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Let K be a knot in an integral homology 3-sphere Y, and Σ the corresponding n-fold cyclic branched cover. Assuming that Σ is a rational homology sphere (which is always the case when n is a prime power), we give a formula for the Lefschetz number of the action that the covering translation induces on the reduced monopole homology of Σ. The proof relies on a careful analysis of the Seiberg--Witten equations on 3-orbifolds and of various η-invariants. We give several applications of our formula: (1) we calculate the Seiberg--Witten and Furuta--Ohta invariants for the mapping tori of all semi-free actions of Z/n on integral homology 3-spheres; (2) we give a novel obstruction (in terms of the Jones polynomial) for the branched cover of a knot in S3 being an L-space; (3) we give a new set of knot concordance invariants in terms of the monopole Lefschetz numbers of covering translations on the branched covers.