$SU(2)$-representations of Branched Covers
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arXiv
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| Format: | Preprint |
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2025
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| author | Ghosh, Sudipta Li, Zhenkun Pinzón-Caicedo, Juanita |
| author_facet | Ghosh, Sudipta Li, Zhenkun Pinzón-Caicedo, Juanita |
| contents | We study the existence of irreducible $SU(2)$-representations for cyclic branched covers of knots in $S^3$. Our main result establishes that if $K$ is a non-trivial prime knot and $d$ is an integer such that $d \geq 2$ and $Σ_d(K)$ is an integer homology sphere, then $π_1(Σ_d(K))$ 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 $d\times d$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_19669 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $SU(2)$-representations of Branched Covers Ghosh, Sudipta Li, Zhenkun Pinzón-Caicedo, Juanita Geometric Topology 57M12, 57R58, 57K18 We study the existence of irreducible $SU(2)$-representations for cyclic branched covers of knots in $S^3$. Our main result establishes that if $K$ is a non-trivial prime knot and $d$ is an integer such that $d \geq 2$ and $Σ_d(K)$ is an integer homology sphere, then $π_1(Σ_d(K))$ 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 $d\times d$ 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. |
| title | $SU(2)$-representations of Branched Covers |
| topic | Geometric Topology 57M12, 57R58, 57K18 |
| url | https://arxiv.org/abs/2508.19669 |