The FKB invariant is the 3d index

Stavros Garoufalidis; Roland van der Veen

doi:10.4171/qt/171

The FKB invariant is the 3d index

Stavros Garoufalidis, Roland van der Veen

Dynamical Systems, Geometry and Mathematical Physics

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Abstract

We identify the q-series associated to an 1-efficient ideal triangulation of a cusped hyperbolic 3-manifold by Frohman and Kania-Bartoszynska with the 3D-index of Dimofte– Gaiotto–Gukov. This implies the topological invariance of the q-series of Frohman and KaniaBartoszynska for cusped hyperbolic 3-manifolds. Conversely, we identify the tetrahedron index of Dimofte–Gaiotto–Gukov as a limit of quantum 6j-symbols.

Original language	English
Pages (from-to)	525-538
Number of pages	14
Journal	Quantum Topology
Volume	13
Issue number	3
DOIs	https://doi.org/10.4171/qt/171
Publication status	Published - 2022

Keywords

3-manifolds
3D-index
ideal tetrahedron
Ideal triangulations
normal surfaces
quantum 6j-symbols
spines
tetrahedron index
TQFT
Turaev–Viro invariants

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title = "The FKB invariant is the 3d index",

abstract = "We identify the q-series associated to an 1-efficient ideal triangulation of a cusped hyperbolic 3-manifold by Frohman and Kania-Bartoszynska with the 3D-index of Dimofte– Gaiotto–Gukov. This implies the topological invariance of the q-series of Frohman and KaniaBartoszynska for cusped hyperbolic 3-manifolds. Conversely, we identify the tetrahedron index of Dimofte–Gaiotto–Gukov as a limit of quantum 6j-symbols.",

keywords = "3-manifolds, 3D-index, ideal tetrahedron, Ideal triangulations, normal surfaces, quantum 6j-symbols, spines, tetrahedron index, TQFT, Turaev–Viro invariants",

author = "Stavros Garoufalidis and {van der Veen}, Roland",

note = "Funding Information: The second author wishes to thank the International Mathematics Center at SUSTech University, Shenzhen for their hospitality. The authors wish to thank Banff for inviting them at the conference on Modular Forms and Quantum Knot Invariants in March 2018 where the results were conceived. Publisher Copyright: {\textcopyright} 2023 European Mathematical Society.",

year = "2022",

doi = "10.4171/qt/171",

language = "English",

volume = "13",

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journal = "Quantum Topology",

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T1 - The FKB invariant is the 3d index

AU - Garoufalidis, Stavros

AU - van der Veen, Roland

N1 - Funding Information: The second author wishes to thank the International Mathematics Center at SUSTech University, Shenzhen for their hospitality. The authors wish to thank Banff for inviting them at the conference on Modular Forms and Quantum Knot Invariants in March 2018 where the results were conceived. Publisher Copyright: © 2023 European Mathematical Society.

PY - 2022

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N2 - We identify the q-series associated to an 1-efficient ideal triangulation of a cusped hyperbolic 3-manifold by Frohman and Kania-Bartoszynska with the 3D-index of Dimofte– Gaiotto–Gukov. This implies the topological invariance of the q-series of Frohman and KaniaBartoszynska for cusped hyperbolic 3-manifolds. Conversely, we identify the tetrahedron index of Dimofte–Gaiotto–Gukov as a limit of quantum 6j-symbols.

AB - We identify the q-series associated to an 1-efficient ideal triangulation of a cusped hyperbolic 3-manifold by Frohman and Kania-Bartoszynska with the 3D-index of Dimofte– Gaiotto–Gukov. This implies the topological invariance of the q-series of Frohman and KaniaBartoszynska for cusped hyperbolic 3-manifolds. Conversely, we identify the tetrahedron index of Dimofte–Gaiotto–Gukov as a limit of quantum 6j-symbols.

KW - 3-manifolds

KW - 3D-index

KW - ideal tetrahedron

KW - Ideal triangulations

KW - normal surfaces

KW - quantum 6j-symbols

KW - spines

KW - tetrahedron index

KW - TQFT

KW - Turaev–Viro invariants

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VL - 13

SP - 525

EP - 538

JO - Quantum Topology

JF - Quantum Topology

IS - 3

ER -

The FKB invariant is the 3d index

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