Paper uses Turaev-Viro TQFT to estimate 3-manifold genus.
problem Estimating the Heegaard genus of 3-manifolds.
method Turaev-Viro state sum TQFT and unitary modular category.
result Provides a lower bound for Heegaard genus using TQFT.
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48 results for “Turaev-Viro TQFT”
We show that for every spherical category $\C$ with invertible dimension, the Turaev-Viro TQFT admits a splitting into blocks which come from an HQFT, called the Turaev-Viro HQFT. The Turaev-Viro HQFT has the classifying space $B\grad$ as target space, where $\grad$ is a group obtained from the category $\C$. This cons…
The paper characterizes boundaries in Turaev-Viro TQFTs and Dijkgraaf-Witten theories.
problem Characterizing boundaries in Turaev-Viro TQFTs and Dijkgraaf-Witten theories.
method Identifying explicit boundary locality conditions and proving consistency with state sum models.
result Turaev-Viro and Dijkgraaf-Witten theories with boundary defects admit a state sum description.
In this paper, we examine Kitaev's lattice model for an arbitrary complex, semisimple Hopf algebra. We prove that this model gives the same topological invariants as Turaev-Viro theory. Using the description of Turaev-Viro theory as an extended TQFT, we prove that the excited states of the Kitaev model correspond to Tu…
In this paper we show how one can extend Turaev-Viro invariants, defined for an arbitrary spherical fusion category , to 3-manifolds with corners. We demonstrate that this gives an extended TQFT which conjecturally coincides with the Reshetikhin-Turaev TQFT corresponding to the Drinfeld center . In the present…
We work in the reduced SU(N,K) modular category as constructed recently by Blanchet. We define spin type and cohomological refinements of the Turaev-Viro invariants of closed oriented 3-manifolds and give a formula relating them to Blanchet's invariants. Roberts' definition of the Turaev-Viro state sum is exploited. Fu…
Study how Turaev-Viro invariants change with cabling operations.
problem Understanding how Turaev-Viro invariants vary with cabling operations.
method Utilized the invertibility of a linear operator associated with torus knot cable spaces in Reshetikhin-Turaev SO3 TQFT.
result Showed the Chen-Yang volume conjecture is stable under (p,q)-cabling for coprime p and q.
In this article, we discuss a (2+1)-dimensional topological quantum field theory, for short TQFT, with a Verlinde basis. As a conclusion of this general theory, we have a Dehn surgery formula. We show that Turaev-Viro-Ocneanu TQFT has a Verlinde basis. Several applications of this theorem are exposed. Based on Izumi's …
In a previous work arXiv:0903.4512, we have built an homotopical Turaev-Viro invariant and an HQFT from the universal graduation of a spherical category. In the present paper, we show that every graduation of a spherical category $\C$ defines an homotopical Turaev-Viro invariant $HTV_{\C}^{(G,p)}$ and an HQFT $…
In this paper, we establish the general theory of (2+1)-dimensional topological quantum field theory (in short, TQFT) with a Verlinde basis. It is a consequence that we have a Dehn surgery formula for 3-manifold invariants for this kind of TQFT's. We will show that Turaev-Viro-Ocneanu unitary TQFT's obtained from subfa…
Extends string-net theory to 3D TQFT via surface graphs and surgery.
problem Formulate 3D TQFT using string-net theory.
method Extend string-net construction to 3D TQFT using surface graphs and surgery.
result Alternative description of Turaev-Viro model using string-nets.
Develops Hermitian TQFTs from quantum groups, defining new topological phases.
problem Defining Hermitian non-semisimple TQFTs.
method Categorical context and representation theory of quantum groups.
result New pseudo-Hermitian topological phases from quantum group representations.
Overview of 3D TQFTs and 3-manifold invariants.
problem Quantum invariants of 3-manifolds.
method Recall and review of TQFTs, fusion categories, and recent generalizations.
result Overview of various 3D TQFTs and their invariants.
A universal state summath.QA
Defines a universal state sum construction for various TQFTs.
problem No specific problem stated; universal construction for TQFTs.
method Defines a universal state sum construction using n-categories with specific conditions.
result Produces state sums from n-categories and handle decompositions of n+1-manifolds.
Pseudo-Hermitian Levin-Wen models from non-semisimple TQFTscond-mat.str-el
New pseudo-Hermitian models from non-semisimple TQFTs.
problem Constructing exactly solvable pseudo-Hermitian spin Hamiltonians.
method Identifying ground states on surfaces using non-semisimple TQFTs.
result Ground states depend only on spatial topology and can be assigned by non-semisimple TQFTs.
In this paper, we establish a rigorous correspondence between the two tube algebras, that one comes from the Turaev-Viro-Ocneanu TQFT introduced by Ocneanu and another comes from the sector theory introduced by Izumi, and construct a canonical isomorphism between the centers of the two tube algebras, which is a conjuga…
The paper defines new TQFTs from non-semisimple categories and proves spherical categories are chromatic.
problem Defining non-compact TQFTs from non-semisimple categories.
method Introducing admissible skein modules, chromatic categories, and using Juhász's cobordism presentation.
result Non-compact (2+1)-TQFTs can be defined from chromatic categories, extending Turaev-Viro TQFTs.
TQFT invariants are either easy or hard to compute, depending on the TQFT type.
problem Computing TQFT invariants on closed 3-manifolds.
method Application of a dichotomy result for weighted constraint satisfaction problems over C.
result TQFT invariants are either solvable in polynomial time or -hard.
Let C be a spherical fusion category. We prove that the Turaev-Viro-Barrett-Westbury state sum invariant of 3-manifolds derived from C is equal to the Reshetikhin-Turaev surgery invariant of 3-manifolds derived from Z(C), where Z(C) is the Drinfeld-Joyal-Street center of C.
It has been conjectured that every -TQFT is a Chern-Simons-Witten (CSW) theory labelled by a pair , where is a compact Lie group, and a cohomology class. We study two TQFTs constructed from Jones' subfactor theory which are believed to be counterexamples to this conjecture: one is the…
Given a TQFT in dimension d+1, and an infinite cyclic covering of a closed (d+1)-dimensional manifold M, we define an invariant taking values in a strong shift equivalence class of matrices. The notion of strong shift equivalence originated in R. Williams' work in symbolic dynamics. The Turaev-Viro module associated to…
By applying a variant of the TQFT constructed by Blanchet, Habegger, Masbaum, and Vogel, and using a construction of Ohtsuki, we define a module endomorphism for each knot K by using a tangle obtained from a surgery presentation of K. We show that it is strong shift equivalent to the Turaev-Viro endomorphism associated…
We establish a relation between the "large r" asymptotics of the Turaev-Viro invariants and the Gromov norm of 3-manifolds. We show that for any orientable, compact 3-manifold , with (possibly empty) toroidal boundary, is bounded above by a function linear in and whose slope is a positiv…
Paper finds Hempel pairs distinguishable by Turaev-Viro invariants.
problem Surface bundles with non-isomorphic fundamental groups.
method Examines Hempel pairs and their Turaev-Viro invariants.
result Examples of Hempel pairs distinguishable and undistinguishable by Turaev-Viro invariants.
Turaev-Viro invariants match for certain surface bundles.
problem Matching Turaev-Viro invariants for surface bundles.
method Profinite completions and spherical fusion categories.
result Invariants coincide for specific surface groups.
The paper finds formulas for a specific invariant order 7.
problem Finding a specific invariant order 7.
method Explicit formulas for a Turaev-Viro invariant of order 7.
result Explicit formulas for the invariant of order 7.
Study Turaev-Viro invariants of Seifert fibered 3-manifolds, proving volume conjecture.
problem Understanding the asymptotic behavior of Turaev-Viro invariants for Seifert fibered 3-manifolds.
method Analysis of large asymptotic behavior of Turaev-Viro invariants.
result Proved the volume conjecture for Seifert fibered 3-manifolds with empty and non-empty boundaries.
The paper proves an asymptotic additivity of Turaev-Viro invariants for a family of 3-manifolds.
problem Preserving the Turaev-Viro invariant volume conjecture under gluings of toroidal boundary components.
method Using a construction of hyperbolic cusped 3-manifolds by Agol, the authors show that the asymptotics of Turaev-Viro invariants are additive under certain gluings of elementary pieces.
result The Turaev-Viro invariant volume conjecture is preserved under specific gluings of 3-manifolds.
A relative version of the Turaev-Viro invariants and the volume of hyperbolic polyhedral 3-manifoldsmath.GT
Defines a new version of Turaev-Viro invariants for 3-manifolds with boundaries.
problem Computing the volume of hyperbolic polyhedral 3-manifolds.
method Introduces a relative version of Turaev-Viro invariants for ideally triangulated compact 3-manifolds with boundaries and a coloring on edges.
result Proves the Volume Conjecture for these invariants, suggesting a method to solve the conjecture for hyperbolic 3-manifolds with totally geodesic boundary.
Study Turaev-Viro invariants of 3-manifolds with toroidal boundary.
problem Volume conjecture for Seifert fibered and graph 3-manifolds.
method Large asymptotic behavior of Turaev-Viro invariants under gluing operation.
result Volume conjecture proven for Seifert fibered and graph 3-manifolds.
We consider certain invariants of links in 3-manifolds, obtained by a specialization of the Turaev-Viro invariants of 3-manifolds, that we call colored Turaev-Viro invariants. Their construction is based on a presentation of a pair (M,L), where M is a closed oriented 3-manifold and L is an oriented link in M, by a tria…
New construction of Turaev-Viro invariants invariant under Morita equivalence.
problem Constructing Turaev-Viro invariants invariant under Morita equivalence.
method Pivotal bicategory construction of spherical module categories.
result The invariant recovers the standard Turaev-Viro invariant and is independent of the skeleton.
The Chen-Yang volume conjecture states that the growth rate of the Turaev-Viro invariants of a compact oriented -manifold determines its simplicial volume. In this paper we prove that the Chen-Yang conjecture is stable under -cabling.
New invariants from quantum group theory for hyperbolic 3-manifolds.
problem Computing invariants for hyperbolic 3-manifolds with boundary.
method Using modular doubles of quantum and -symbols.
result Invariants decay exponentially with hyperbolic volume and 1-loop terms.
Turaev Viro invariants are amongst the most powerful tools to distinguish 3-manifolds: They are implemented in mathematical software, and allow practical computations. The invariants can be computed purely combinatorially by enumerating colourings on the edges of a triangulation T. These edge colourings can be interpre…
Asymptotic Behavior of Colored Jones polynomial and Turaev-Viro Invariant of figure eight knotmath.GT
In this paper we investigate the asymptotic behavior of the colored Jones polynomials and the Turaev-Viro invariants for the figure eight knot. More precisely, we consider the -th colored Jones polynomials evaluated at -th root of unity with a fixed limiting ratio, , of and . We find out the…
We obtain a formula for the Turaev-Viro invariants of a link complement in terms of values of the colored Jones polynomial of the link. As an application we give the first examples for which the volume conjecture of Chen and the third named author\,\cite{Chen-Yang} is verified. Namely, we show that the asymptotics of t…
The Turaev-Viro invariants are a powerful family of topological invariants for distinguishing between different 3-manifolds. They are invaluable for mathematical software, but current algorithms to compute them require exponential time. The invariants are parameterised by an integer . We resolve the question …
Study non-semisimple TQFT for Burau representation density and unitarity.
problem Density and unitarity of the Burau representation from a non-semisimple TQFT perspective.
method TQFT construction of Squier's Hermitian form on the Burau representation.
result Density of the image of braid group in unitary representations.
The article calculates asymptotic expansions for quantum invariants from surgeries on Whitehead link components.
problem Calculating quantum invariants for 3-manifolds resulting from surgeries on Whitehead link components.
method Asymptotic expansion of relative Reshetikhin-Turaev and Turaev-Viro invariants.
result Asymptotic formulas for both invariants are derived.
We give a categorical setting in which Penrose graphical calculus naturally extends to graphs drawn on the boundary of a handlebody. We use it to introduce invariants of 3-manifolds presented by Heegaard splittings. We recover Kuperberg invariants when the category comes from an involutory Hopf algebra and Turaev-Viro …
Classifies 2D TQFTs for orientable cobordisms using additional data.
problem Classifying 2D TQFTs for orientable cobordisms.
method Describes an intermediate framework using an involution and Möbius strip value.
result Intermediate classification of 2D TQFTs for orientable cobordisms.
Proves Witten-Reshetikhin-Turaev 3-TQFT as a boundary condition of Crane-Yetter 4-TQFT.
problem Proving a boundary condition for Crane-Yetter 4-TQFT.
method Extending ideas of Crane-Yetter and Jordan, proving Crane-Yetter 4-TQFT and its non-semisimple version are once-extended TQFTs, defining a boundary condition.
result Reconstructs Witten-Reshetikhin-Turaev 3-TQFT and its non-semisimple versions using Crane-Yetter 4-TQFT.
This paper categorifies Quinn's TQFTs and computes them for specific omega-groupoids.
problem Constructing and computing finite total homotopy TQFTs.
method Direct homotopy theoretical construction, categorification of Quinn's TQFTs, explicit computation for omega-groupoids.
result Categorification and explicit computation of Quinn's TQFTs for omega-groupoids.
Defines extended TQFTs using handle attachments.
problem Constructing extended topological quantum field theories (TQFTs).
method Finite presentation of cobordism symmetric monoidal bicategory using handle attachments and relations.
result Constructs a once extended TQFT from categorified TQFT and handle 2-morphisms.
New (3+1) TQFTs created from non-semisimple categories.
problem Creating TQFTs from non-semisimple ribbon categories.
method Using skein theory and admissible skein modules, defining TQFTs with specific algebraic conditions.
result Explicit realization of a TQFT based on the cobordism hypothesis.
Almost integral TQFTs were introduced by Gilmer [Duke Math. J. 125 (2004) 389--413]. The aim of this paper is to modify the TQFT of the category of extended 3-cobordisms given by Turaev (in his book: Quantum invariants of knots and 3-manifolds) to obtain an almost integral TQFT.
Chern-Simons and Reshetikhin-Turaev theories are shown equivalent for U(1) gauge group.
problem Equivalence between Chern-Simons and Reshetikhin-Turaev TQFTs.
method Proof of natural isomorphism between theories for finite quadratic modules.
result Extended -dimensional TQFTs are naturally isomorphic.