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Singular Loci of SL2(C)-Representation Varieties of Torus Knot Groups
Abstract
Let T(q,p)=⟨ x,y| xp=yq⟩, where p,q>1 are relatively prime, and let R=Hom(T(q,p),SL2(ℂ)). We compute the integral homology of the singular locus of R. We show that this locus has complex dimension $2$ and is a disjoint union of μ=(p−1)(q−1) smooth affine surfaces, each homotopy equivalent to the two-dimensional sphere. Consequently, H0(Sing R;Z)≅ H2(Sing R;Z)≅Zμ, and all remaining homology groups vanish. Thus both nonzero Betti numbers equal the first Betti number of the Milnor fiber of the plane curve singularity zp+wq=0, namely its Milnor number μ.
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BibTeX
@misc{liriano2026singular,
title = {{Singular Loci of SL2(C)-Representation Varieties of Torus Knot Groups}},
author = {S. Liriano},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32064},
eprint = {2610.32064},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32064},
}
RIS
TY - UNPB
TI - Singular Loci of SL2(C)-Representation Varieties of Torus Knot Groups
AU - S. Liriano
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32064
UR - https://arxiv.org/abs/2610.32064
AB - Let T(q,p)=⟨ x,y| x^p=y^q⟩, where p,q>1 are relatively prime, and let R=Hom(T(q,p),SL_2(ℂ)). We compute the integral homology of the singular locus of R. We show that this locus has complex dimension $2$ and is a disjoint union of μ=(p−1)(q−1) smooth affine surfaces, each homotopy equivalent to the two-dimensional sphere. Consequently, H_0(Sing R;Z)≅ H_2(Sing R;Z)≅Z^μ, and all remaining homology groups vanish. Thus both nonzero Betti numbers equal the first Betti number of the Milnor fiber of the plane curve singularity z^p+w^q=0, namely its Milnor number μ.
ER -