# 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}
}
• Published 31 July 2020
• Mathematics
• arXiv: Geometric Topology
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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## References

SHOWING 1-10 OF 56 REFERENCES

### Witten–Reshetikhin–Turaev Invariants of¶Seifert Manifolds

• Mathematics
• 1999
Abstract:For Seifert homology spheres, we derive a holomorphic function of K whose value at integer K is the sl2 Witten–Reshetikhin–Turaev invariant, ZK, at q= exp 2πi/K. This function is expressed

### Reshetikhin-Turaev invariants of Seifert 3-manifolds for classical simple Lie algebras, and their asymptotic expansions

• Mathematics
• 2002
We derive formulas for the Reshetikhin-Turaev invariants of all oriented Seifert manifolds associated to an arbitrary complex finite dimensional simple Lie algebra $\mathfrak g$ in terms of the

### ON THE QUANTUM INVARIANT FOR THE BRIESKORN HOMOLOGY SPHERES

We study an exact asymptotic behavior of the Witten–Reshetikhin–Turaev SU(2) invariant for the Brieskorn homology spheres Σ(p1, p2, p3) by use of properties of the modular form following a method

### Quantum invariants, modular forms, and lattice points II

We study the SU(2) Witten-Reshetikhin-Turaev (WRT) invariant for the Seifert fibered homology spheres with M-exceptional fibers. We show that the WRT invariant can be written in terms of

### Quantum invariant, modular form, and lattice points

We study the Witten--Reshetikhin--Turaev SU(2) invariant for the Seifert manifold with 4-singular fibers. We define the Eichler integrals of the modular forms with half-integral weight, and we show

### Fivebranes and 3-manifold homology

• Mathematics
• 2016
A bstractMotivated by physical constructions of homological knot invariants, we study their analogs for closed 3-manifolds. We show that fivebrane compactifications provide a universal description of

### Localization for Wilson Loops in Chern-Simons Theory

We reconsider Chern-Simons gauge theory on a Seifert manifold M , which is the total space of a nontrivial circle bundle over a Riemann surface Σ, possibly with orbifold points. As shown in previous

### Quantum invariants of Seifert 3–manifolds and their asymptotic expansions

• Mathematics
• 2002
We report on recent results of the authors concerning calculations of quantum invariants of Seifert 3–manifolds. These results include a derivation of the Reshetikhin–Turaev invariants of all

### Resurgent Analysis for Some 3-manifold Invariants

We study resurgence for some 3-manifold invariants when $G_{\mathbb{C}}=SL(2, \mathbb{C})$. We discuss the case of an infinite family of Seifert manifolds for general roots of unity and the case of

### Resurgence analysis of quantum invariants of Seifert fibered homology spheres

• Mathematics
Journal of the London Mathematical Society
• 2022
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