Researchers found the -invariant for is constant regardless of .
problem Exploring the -invariant for quotient groups .
method Analyzing the -invariant for and extending to .
result The -invariant for is independent of .
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26 results for “Z-invariants”
Nesting behind -invariantsmath.RT
Proposes categorification of Z-invariants for specific 3-manifolds.
problem Categorification of Z-invariants for negative definite plumbed 3-manifolds.
method Abelian categorification using 3d N=2 theory and log VOAs.
result Nested Weyl-type character formulas reconstruct -invariants.
Witt invariants from q-series math.GT
New relation found between 3-manifold Witt invariants and -invariants.
problem Computing Witt invariants of 3-manifolds.
method Using -invariants and analyzing homology spheres.
result Alternative method to compute Witt invariants.
Develops a TQFT framework to compute invariants of three-manifolds.
problem Understanding the TQFT structure of invariants of three-manifolds.
method Decorated Spin-TQFTs, novel quantization of Chern-Simons theory, and algebra of observables.
result Explicit closed-form expressions for invariants of various three-manifolds.
3d Modularity Revisitedhep-th
Researchers clarify modular group representations and vertex operator algebras for 3d invariants.
problem Understanding the full set of 3d invariants and their modular properties.
method Introducing supersymmetric defects and constructing cone vertex operator algebras.
result The full vector-valued quantum modular form for \(\widetilde{
m SL}_2(\mathbb{Z})\) captures all \(\hat Z\)-invariants of a given three-manifold.
Study Alexander polynomials of modular knots, revealing finite and infinite coefficient properties.
problem Investigate Alexander polynomials of modular knots.
method Use Burau representation and geometric -invariants.
result Alexander polynomials of modular knots have both finite and infinite coefficient properties.
We prove that the well-rounded retract of SO_n\SL_n(R) is a minimal SL_n(Z)-invariant spine.
Constructs 3D topological field theories from a specific quantum group, linking to physics invariants.
problem Developing topological field theories from non-semisimple quantum groups.
method Using the unrolled quantum group of and a relative modular structure on weight modules.
result Establishes a connection between constructed invariants and physicists' -invariants.
Abstract studies 3-manifolds and vertex algebras, expanding known connections.
problem Understanding the relationship between 3-manifolds and vertex algebras.
method Developed a dictionary between 3-manifolds and vertex algebras using -series.
result Generalized known entries to higher rank Lie groups and 3-manifolds with toral boundaries.
New invariant for plumbed 3-manifolds with detailed computations and conjectures.
problem Computing and understanding invariants for plumbed 3-manifolds.
method Introducing a two-variable refinement , analytically computing limits, and proposing conjectures based on numerical data.
result Proposed conjecture that the recovered is an invariant for all tree plumbed 3-manifolds.
The groups of differential characters of Cheeger and Simons admit a natural multiplicative structure. The map given by the squares of degree 2k differential characters reduces to a homomorphism of ordinary cohomology groups. We prove that the homomorphism factors through the Steenrod squaring operation of degree 2k. A …
Study of origamis in minimal stratum with single cylinders, calculating spin parities and monodromy groups.
problem Understanding the structure and properties of origamis in the minimal stratum of moduli space.
method Construction and analysis of minimal -origamis, calculation of spin parities, and investigation of monodromy groups.
result All minimal -origamis have monodromy groups that are almost always finite simple groups.
Study compares two methods to extend invariants, finding incompatibility for Brieskorn spheres.
problem Comparing two methods to extend invariants for 3-manifolds.
method Two prescriptions: regularized -surgery combined with false-mock modular conjecture, and resurgence-based construction.
result Incompatibility found between the two prescriptions for some Brieskorn spheres.
We study tori which are cyclic covers of the standard torus, that is, the deck transformation group of the covering map is cyclic. These covering tori can be parametrized in a natural way and we show that being cyclic is equivalent to certain arithmetic condition on these parameters. There is a natural $\mathrm{SL}(2,\…
Researchers derive -series for and groups.
problem Deriving -series for and groups.
method Change of variable relating link invariants to and link invariants.
result Explicit -series for and groups.
New invariant unifies two theories of 3-manifolds, recovering quantum invariants.
problem Quantum invariants of 3-manifolds at roots of unity.
method Introduces a unified invariant for negative definite plumbed 3-manifolds.
result 2-variable refinement of the -invariant.
For a closed smooth manifold admitting a symplectic structure, we define a smooth topological invariant using almost-Kähler metrics, i.e. Riemannian metrics compatible with symplectic structures. We also introduce depending on symplectic deformation equivalence class . We first prove tha…
The thesis examines the relationship between three-manifold invariants and knot theory, finding equalities and patterns.
problem Examining the relationship between three-manifold invariants and knot theory.
method Analytic continuation and quiver representation theory.
result Found equalities and patterns in knot theory and quiver representation.
Study cohomology of GL₂n(Z) and graph complexes using Pfaffian forms.
problem Cohomology of GL₂n(Z) and related graph complexes.
method Use Pfaffian forms on symmetric spaces and dual Laplacians on graphs.
result First cocycle gives a non-trivial class in H⁻⁶(GC₃).
This work reconstructs knot invariants from Alexander polynomials, proving consistency with known theorems.
problem Reconstructing knot invariants from Alexander polynomials.
method Quantization, deformation, and rewriting of Alexander polynomials.
result Derives new formulae for colored superpolynomials and proves consistency with Melvin-Morton-Rozansky theorem.
New invariants for 3-manifolds derived from supergroup representations.
problem Developing invariants for 3-manifolds using supergroup analogues.
method Introducing supergroup analogues of 3-manifold invariants for superunitary groups, focusing on SU(2|1). Calculating q-series for specific 3-manifolds and studying their properties.
result Explicit calculation and study of q-series for certain 3-manifolds, providing a formula relating new invariants to quantum invariants.
New invariant defined for unoriented knots, proving no factorization through topological concordance.
problem Defining and proving properties of unoriented slice-torus invariants.
method Introducing and proving properties of unoriented slice-torus invariants.
result Unoriented slice-torus invariants do not factor through the topological concordance group.
We study 4d superconformal indices for a large class of N=1 superconformal quiver gauge theories realized combinatorially as a bipartite graph or a set of "zig-zag paths" on a two-dimensional torus T^2. An exchange of loops, which we call a "double Yang-Baxter move", gives the Seiberg duality of the gauge theory, and t…
Quantum modularity proved for a knot manifold.
problem Proving quantum modularity for a specific closed hyperbolic 3-manifold.
method Using factorization of state integrals and proving quantum modularity for functions and -series.
result Quantum modularity for the closed manifold provides a unification of volume conjecture and Witten's asymptotic expansion conjecture.
The current article stems from our study on the asymptotic behavior of holomorphic isometric embeddings of the Poincaré disk into bounded symmetric domains. As a first result we prove that any holomorphic curve exiting the boundary of a bounded symmetric domain must necessarily be asymptotically totally geodesic. A…
Torelli groups' homology is finitely generated in stable range.
problem Whether the homology groups of Torelli subgroups are finitely generated in stable range.
method Using unipotency condition and Tavgen's theorem.
result Homology groups are finitely generated in stable range.