2025/08/27 by Ghosh, Sudipta, Li, Zhenkun, Pinzón-Caicedo, Juanita
#57K18 #57M12 #57R58 #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2508.19669
We study the existence of irreducible SU(2)-representations for cyclic branched covers of knots in S3. Our main result establishes that if K is a non-trivial prime knot and d is an integer such that d ≥ 2 and Σd(K) is an integer homology sphere, then π1(Σd(K)) admits an irreducible SU(2)-representation, whenever K satisfies one of two conditions: either K is 2-periodic, or K can be represented as the closure of a tangle adapted to a d× d SICUP matrix. The first condition leverages a commuting trick for covering spaces to realize higher-degree branched covers as 2-fold covers, allowing us to apply recent results of Kronheimer-Mrowka and others. The second condition uses equivariant surgery descriptions and the ν^\sharp invariant from instanton Floer homology. As applications, we provide new infinite families of hyperbolic integer homology spheres admitting irreducible representations, including examples where previously known criteria fail.