Authors: | Ekholm, Tobias Gruen, Angus Gukov, Sergei Kucharski, Piotr Park, Sunghyuk Stošić, Marko Sułkowski, Piotr |

Affiliations: | Mathematics Mathematical Institute of the Serbian Academy of Sciences and Arts |

Title: | Branches, quivers, and ideals for knot complements |

Journal: | Journal of Geometry and Physics |

Volume: | 177 |

First page: | 104520 |

Issue Date: | 1-Jul-2022 |

Rank: | ~M22 |

ISSN: | 0393-0440 |

DOI: | 10.1016/j.geomphys.2022.104520 |

Abstract: | We generalize the FK invariant, i.e. Zˆ for the complement of a knot K in the 3-sphere, the knots-quivers correspondence, and A-polynomials of knots, and find several interconnections between them. We associate an FK invariant to any branch of the A-polynomial of K and we work out explicit expressions for several simple knots. We show that these FK invariants can be written in the form of a quiver generating series, in analogy with the knots-quivers correspondence. We discuss various methods to obtain such quiver representations, among others using R-matrices. We generalize the quantum a-deformed A-polynomial to an ideal that contains the recursion relation in the group rank, i.e. in the parameter a, and describe its classical limit in terms of the Coulomb branch of a 3d-5d theory. We also provide t-deformed versions. Furthermore, we study how the quiver formulation for closed 3-manifolds obtained by surgery leads to the superpotential of 3d N=2 theory T[M3] and to the data of the associated modular tensor category MTC[M3]. |

Keywords: | A polynomial | Open curve counts | Quantum invariants; High Energy Physics - Theory; High Energy Physics - Theory; Mathematics - Geometric Topology; Mathematics - Quantum Algebra; Mathematics - Symplectic Geometry |

Publisher: | Elsevier |

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