Witten–Reshetikhin–Turaev Function for a Knot in Seifert Manifolds

@article{Fuji2020WittenReshetikhinTuraevFF,
  title={Witten–Reshetikhin–Turaev Function for a Knot in Seifert Manifolds},
  author={Hiroyuki Fuji and Kohei Iwaki and Hitoshi Murakami and Yuji Terashima},
  journal={arXiv: Geometric Topology},
  year={2020}
}
In this paper, for a Seifert loop (i.e., a knot in a Seifert three-manifold), first we give a family of an explicit function $\Phi(q; N)$ whose special values at roots of unity are identified with the Witten-Reshetikhin-Turaev invariants of the Seifert loop for the integral homology sphere. Second, we show that the function $\Phi(q; N)$ satisfies a $q$-difference equation whose classical limit coincides with a component of the character varieties of the Seifert loop. Third, we give an… 

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For a Seifert fibered homology sphere X$X$ , we show that the q$q$ ‐series invariant Ẑ0(X;q)$\hat{\operatorname{Z}}_0(X;q)$ , introduced by Gukov–Pei–Putrov–Vafa, is a resummation of the Ohtsuki
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