Proves a quantum invariant conjecture for specific three-manifolds.
problem Proving a conjecture about quantum invariants for certain three-manifolds.
method Gives an integral representation and resurgent asymptotic expansion for the average of GPPV invariants.
result Proves the Costantino--Geer--Patureau-Mirand invariant for negative definite plumbed three-manifolds.
Special shadow-complexity equals k+1 for k copies of S1×S3.
problem Calculating the special shadow-complexity of a specific 4-manifold.
method Defined by Costantino using Turaev's shadows, proved for connected sums of S1×S3.
result The special shadow-complexity of k copies of S1×S3 is k+1.
We develop the general theory for the construction of Extended Topological Quantum Field Theories (ETQFTs) associated with the Costantino-Geer-Patureau quantum invariants of closed 3-manifolds. In order to do so, we introduce relative modular categories, a class of ribbon categories which are modeled on representations…
Researchers compute TQFT representation for sphere with 4 punctures.
problem Computing the representation of mapping class group for a sphere with 4 punctures.
method Non semi-simple TQFT approach, focusing on sphere with 4 punctures.
result The representation is faithful and compared with braid groups.
This survey covers some of the results contained in the papers by Costantino, Geer and Patureau (https://arxiv.org/abs/1202.3553) and by Blanchet, Costantino, Geer and Patureau (https://arxiv.org/abs/1404.7289). In the first one the authors construct two families of Reshetikhin-Turaev-type invariants of 3-manifolds, $\…
New invariant from Viro's gl(1|1) polynomial distinguishes lens spaces.
problem Constructing a 3-manifold invariant from Viro's gl(1|1) polynomial.
method Following Costantino, Geer, and Patureau-Mirand's method in relative G-modular categories.
result The invariant can distinguish homotopy equivalent lens spaces.
Study compares WRT and CGP invariants using Habiro's series.
problem Comparing WRT and CGP invariants for knots.
method Established relationship between Habiro's series and ADO invariants.
result Difference between WRT and CGP invariants determined by Habiro series.
Paper proves Reshetikhin-Turaev link invariants appear in higher order terms of re-normalized link invariants for plumbed links.
problem Proving relations between Reshetikhin-Turaev and re-normalized link invariants for links.
method Analyzing higher order terms of re-normalized link invariants for plumbed links.
result Reshetikhin-Turaev link invariants appear in higher order terms of re-normalized link invariants for plumbed links.
Constructs TQFTs for cobordisms with cohomology class decorations.
problem Creating TQFTs for cobordisms with cohomology class decorations.
method Starting from an abelian group G and a factorizable ribbon Hopf G-bialgebra H, constructs a TQFT JH for connected framed cobordisms between connected surfaces with connected boundary decorated with cohomology classes with coefficients in G. result Our functor recovers a special case of Kerler-Lyubashenko TQFTs when restricted to trivial decorations.
The paper proves a combinatorial Ricci flow converges to hyperbolic structures on certain 3-manifolds.
problem Proving convergence of combinatorial Ricci flow to hyperbolic structures.
method Combinatorial Ricci flow on closed pseudo 3-manifolds with specific edge valences.
result Existence and uniqueness of a complete hyperbolic metric with totally geodesic boundary.
Relates two types of skein algebras using explicit correspondences.
problem Defining and relating stated and internal skein algebras.
method Explicit correspondence between stated and internal skein algebras, distinguishing between left and right boundary edges, proving excision properties.
result Agrees with excision properties of stated skein algebras under specific conditions.
Researchers compare two methods for handlebody constructions, finding they are related with a 'background charge'.
problem Comparing two methods for handlebody constructions in finite ribbon categories.
method Admissible skein module construction vs. ansular functor construction.
result An isomorphism between the two constructions is proven, with a 'background charge' that becomes trivial in the unimodular case.
The paper connects 3D manifold invariants to hyperbolic cone metrics and discrete Fourier transforms.
problem Volume conjecture for Reshetikhin-Turaev invariants of 3-manifolds with links.
method Volume conjecture, hyperbolic cone metrics, discrete Fourier transforms, change-of-pair operations.
result Volume conjecture proven for specific cases, provides approach to solving Volume Conjecture for hyperbolic 3-manifolds.