The 3D index of an ideal triangulation and angle structures

Stavros Garoufalidis

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The 3D index of an ideal triangulation and angle structures

Stavros Garoufalidis

Geometric Topology · Submitted 8 Aug 2012

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Our goal is to give an axiomatic definition of the tetrahedron index, and a proof that the domain of the 3D index consists precisely of the set of ideal triangulations that support an index structure.

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.81 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.10 · compares with prior results 0.26 · mentions code or data availability 0.02.

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arXiv listing abstract v2. Results and the main claim are underlined; show every role.

The 3D index of Dimofte-Gaiotto-Gukov a partially defined function on the set of ideal triangulations of 3-manifolds with $r$ torii boundary components. For a fixed $2r$ tuple of integers, the index takes values in the set of $q$-series with integer coefficients. Our goal is to give an axiomatic definition of the tetrahedron index, and a proof that the domain of the 3D index consists precisely of the set of ideal triangulations that support an index structure. The latter is a generalization of a strict angle structure. We also prove that the 3D index is invariant under 3-2 moves, but not in general under 2-3 moves.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The 3D index of Dimofte-Gaiotto-Gukov a partially defined function on the set of ideal triangulations of 3-manifolds with $r$ torii boundary components.	background 1.00	0.06	0.05	asserted 0.99
s1	For a fixed $2r$ tuple of integers, the index takes values in the set of $q$-series with integer coefficients.	background 0.93	0.18	0.05	asserted 1.00
s2	Our goal is to give an axiomatic definition of the tetrahedron index, and a proof that the domain of the 3D index consists precisely of the set of ideal triangulations that support an index structure.	contribution 0.99	0.84	0.50	asserted 0.92
s3	The latter is a generalization of a strict angle structure.	background 0.42	0.26	0.06	asserted 0.99
s4	We also prove that the 3D index is invariant under 3-2 moves, but not in general under 2-3 moves.	result 0.99	0.93	0.19	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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The paper

Original abstract

The 3D index of Dimofte-Gaiotto-Gukov a partially defined function on the set of ideal triangulations of 3-manifolds with $r$ torii boundary components. For a fixed $2r$ tuple of integers, the index takes values in the set of $q$-series with integer coefficients. Our goal is to give an axiomatic definition of the tetrahedron index, and a proof that the domain of the 3D index consists precisely of the set of ideal triangulations that support an index structure. The latter is a generalization of a strict angle structure. We also prove that the 3D index is invariant under 3-2 moves, but not in general under 2-3 moves.

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References

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

RG Domain Walls and Hybrid Triangulations
hep-th 2013-04 conditional novelty 8.0 of 10

Every three-dimensional cobordism yields a three-dimensional Chern-Simons-matter theory that serves as a domain wall between four-dimensional N=2 gauge theories, including a new class of renormalization-group walls.
Complex Chern-Simons theory at level k via the 3d-3d correspondence
hep-th 2014-09 accept novelty 7.0 of 10

A state-integral model is constructed that computes SL(n,C) Chern-Simons partition functions at arbitrary level k from ideal triangulations, generalizing the known k=0 and k=1 cases.
K-Decompositions and 3d Gauge Theories
hep-th 2013-01 conditional novelty 7.0 of 10

A hypersimplicial K-decomposition of ideal triangulations yields octahedron gluing equations that construct framed flat PGL(K,C) moduli spaces and candidate 3d N=2 theories T_K[M].

How the literature uses it

21 arXiv papers cite this in 38 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 28 sentences without a confident label

Builds on it 1 paper

We discuss the 3D index with quantum trace map, which we call $ℐ_M+Ô (m,e;q)$ and $ℐ_M_P/Q+Ô (q)$, extending the previous works [ref] .

Quantum trace map for 3-manifolds and a 'length conjecture' · 3D index with quantum trace map · 1.00

Compares with it 1 paper

rem.3D Equation [ref] is reminiscent to the 3D index of a tetrahedron introduced by Dimofte-Gaiotto-Gukov [ref] and further studied by [ref] and [ref] .

A generating series for Murakami-Ohtsuki-Yamada graph evaluations · A generating series for the HOMFLY evaluation of MOY graphs · 0.94

Cites it as support 1 paper

is well defined and only contains positive powers of $q$ for all $(𝔔2α⃗,𝔔2β⃗) ∈[M,t]_n$, [ref] .

Complex Chern-Simons theory at level k via the 3d-3d correspondence · Physical meaning of the angle polytope: R-charges of operators · 0.97

Cites it as background 6 papers

In Section [ref] we review the basic properties of the tetrahedron index from [ref] , and give a detailed discussion of the 3D index for an ideal triangulation of a cusped 3-manifold.

1-efficient triangulations and the index of a cusped hyperbolic 3-manifold · Plan of the paper · 0.94
(See [ref] for the details.)

The 3D-index and normal surfaces · The tetrahedral index · 0.90
Recall that the index of an ideal triangulation was introduced in [ref] , necessary and sufficient conditions for its convergence was established in [ref] and its topological invariance was proven in [ref] .

From state integrals to q-series · The case of the $4_1$ knot · 0.99

What it cites · 8 sentences without a confident label

Uses data or values from 1 paper

Let $$ denote the ideal triangulation [ref] of the 1-cusped census manifold m136 using 7 tetrahedra.

Triangulations of hyperbolic 3-manifolds admitting strict angle structures · An ideal triangulation of m136 · 0.89

Cites as support 1 paper

The latter is a nonempty (generally infinite) set by [ref] , assuming that $M$ satisfies [ref].

Triangulations of hyperbolic 3-manifolds admitting strict angle structures · On the topological invariance of the index · 1.00

Cites as background 7 papers

In a series of papers [ref] , Dimofte-Gaiotto-Gukov studied topological gauge theories using as input an ideal triangulation $$ of a 3-manifold $M$.

Gauge Theories Labelled by Three-Manifolds · Introduction · 1.00
In a series of papers [ref] , Dimofte-Gaiotto-Gukov studied topological gauge theories using as input an ideal triangulation $$ of a 3-manifold $M$.

3-Manifolds and 3d Indices · Introduction · 1.00
Consider two ideal triangulations $$ and $$ with $N$ and $N+1$ tetrahedra, respectively, related by a $2-3$ move [equation] The above figure matches the conventions of [ref] .

The quantum content of the gluing equations · Invariance under the $3→2$ move · 0.82

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