RECORD
Observables in the Turaev-Viro and Crane-Yetter models
Quantum Algebra · Submitted 12 Nov 2004
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Some examples are computed which show that the partition function determines an invariant which can detect non locally-flat surfaces in a four-manifold.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
We define an invariant of graphs embedded in a three-manifold and a partition function for 2-complexes embedded in a triangulated four-manifold by specifying the values of variables in the Turaev-Viro and Crane-Yetter state sum models. In the case of the three-dimensional invariant, we prove a duality formula relating its Fourier transform to another invariant defined via the coloured Jones polynomial. In the case of the four-dimensional partition function, we give a formula for it in terms of a regular neighbourhood of the 2-complex and the signature of its complement. Some examples are computed which show that the partition function determines an invariant which can detect non locally-flat surfaces in a four-manifold.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We define an invariant of graphs embedded in a three-manifold and a partition function for 2-complexes embedded in a triangulated four-manifold by specifying the values of variables in the Turaev-Viro and Crane-Yetter state sum models. | contribution 0.99 | 0.94 | 0.24 | asserted 1.00 |
| s1 | In the case of the three-dimensional invariant, we prove a duality formula relating its Fourier transform to another invariant defined via the coloured Jones polynomial. | result 0.93 | 0.94 | 0.18 | asserted 1.00 |
| s2 | In the case of the four-dimensional partition function, we give a formula for it in terms of a regular neighbourhood of the 2-complex and the signature of its complement. | result 0.87 | 0.94 | 0.19 | asserted 1.00 |
| s3 | Some examples are computed which show that the partition function determines an invariant which can detect non locally-flat surfaces in a four-manifold. | result 0.93 | 0.93 | 0.32 | asserted 0.99 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
We define an invariant of graphs embedded in a three-manifold and a partition function for 2-complexes embedded in a triangulated four-manifold by specifying the values of variables in the Turaev-Viro and Crane-Yetter state sum models. In the case of the three-dimensional invariant, we prove a duality formula relating its Fourier transform to another invariant defined via the coloured Jones polynomial. In the case of the four-dimensional partition function, we give a formula for it in terms of a regular neighbourhood of the 2-complex and the signature of its complement. Some examples are computed which show that the partition function determines an invariant which can detect non locally-flat surfaces in a four-manifold.
Human review
References
How the literature uses it
10 arXiv papers cite this in 27 sentences
Papers citing it · 15 sentences without a confident label
Contests it 1 paper
-
Now we will describe it geometrically in terms of a marked Heegaard diagram.[This construction corrects what appears to be a mistake or typo in [ref] .
Conformal Turaev-Viro Theory · Marked Heegaard Diagrams · contest 0.90
Builds on it 3 papers
-
We will not attempt to characterize the admissible $(M_E, Γ)$, which is an interesting question for the future, but setting aside this caveat, the derivations of the main results in $M_E$ are identical to [ref] .
Conformal Turaev-Viro Theory · More general embedding manifolds · 0.89
-
This extends to manifolds with colored graphs [ref] .
Categorical Center of Higher Genera and 4D Factorization Homology · TQFTs as higher invariants · 0.81
-
This generalizes the analogous relation known in the discrete-spectrum case [ref] .
The many facets of a hyperbolic tetrahedron: open and closed triangulations of 3d gravity · Introduction · 0.81
Compares with it 1 paper
-
The identity [ref] is closely analogous to a known result for discrete spin networks [ref] .
Conformal Turaev-Viro Theory · Introduction · 0.98
Cites it as support 3 papers
-
However, it was proved in [ref] that the amplitude $⟨Γ_CH()⟩$ is independent of the Heegaard splitting.
Conformal Turaev-Viro Theory · Chain mail from a Heegaard splitting · 0.96
-
From the handle-decomposition description [ref] , it is clear that CY and RT theory are indeed directly related.
Simplicity condition and boundary-bulk duality · 4-3-2 · 0.96
-
This picture is also supported by spinfoam studies [ref] where it was shown that the invariants of four dimensional Crane-Yetter model are equivalent to Turaev-Viro invariants on the three dimensional boundary.
Effective particle kinematics from Quantum Gravity · From gravity in $4d$ to $3d$ Chern–Simons theory · 1.00
Cites it as background 2 papers
-
The relation with RT invariants has been extensively studied [ref] .
Simplicity condition and boundary-bulk duality · 4-3-2 · 0.87
-
Indeed, the study of CTV theory in [ref] borrows heavily from the literature on spin networks [ref] , especially Roberts' chain-mail formalism [ref] and its extension by Barrett, Garcia-Islas, and Martins to include what are called external edges below [ref] .
Triangulating quantum gravity in AdS$_3$ · Related approaches · 0.81
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.