Finite image of mapping class group representations proved using graph embeddings.
problem Finiteness of images of mapping class group representations in twisted Dijkgraaf-Witten theory.
method Translation of problem into graph manipulation, using TVBW representations and spherical fusion categories.
result Finiteness of images of mapping class group representations in twisted Dijkgraaf-Witten theory is proven.
In this paper, we provide a construction of a state-sum model for finite gauge-group Dijkgraaf-Witten theory on surfaces with codimension 1 defects. The construction requires not only that the triangulation be subordinate to the filtration, but flag-like: each simplex of the triangulation is either disjoint from the de…
We give an interpretation of Yetter's Invariant of manifolds M in terms of the homotopy type of the function space TOP(M,B(G)), where G is a crossed module and B(G) is its classifying space. From this formulation, there follows that Yetter's invariant depends only on the homotopy type of M, and the weak homot…
We compute the Dijkgraaf-Witten invariants of surfaces in terms of projective representations of groups. As an application we prove that the complex Dijkgraaf-Witten invariants of surfaces of positive genus are positive integers.
This paper calculates Dijkgraaf-Witten invariants from Chern classes.
problem Computing Dijkgraaf-Witten invariants from specific 3-cocycles.
method Defines and computes invariants from second Chern classes of representations.
result Clarifies topological interpretation under certain conditions.
We briefly review 3-dimensional untwisted Dijkgraaf-Witten theory over a finite group Γ, and present a method of computing untwisted Dijkgraaf-Witten invariants for arborescent links. Some explicit formulas are given when Γ=Z/pZ⋊Z/(p−1)Z for an odd prime p.
We derive a formula for the Dijkgraaf-Witten invariants of orientable Seifert 3-manifolds with orientable bases.
Arithmetic Dijkgraaf-Witten theory constructs analogues in Chern-Simons TQFT.
problem Developing arithmetic analogues in Chern-Simons TQFT.
method Constructing arithmetic analogues of Chern-Simons 1-cocycle, prequantization bundle, and Chern-Simons functional.
result Decomposition and gluing formulas for arithmetic Chern-Simons invariants and arithmetic Dijkgraaf-Witten partition functions.
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.
Alexander polynomial constraints derived from Seifert surgeries.
problem Constraints on knot surgery results.
method Dijkgraaf-Witten invariant applied to abelian groups.
result Alexander polynomial constraints on knot surgeries.
New invariants for 3-manifolds using special G-systems.
problem Constructing invariants for 3-manifolds.
method Introducing special G-systems and their cohomological construction.
result Simple one-dimensional special G-systems can be constructed using group cohomologies.
The paper extends arithmetic Chern-Simons invariants to real quadratic fields and calculates mod 2 Dijkgraaf-Witten invariants.
problem Calculating arithmetic Dijkgraaf-Witten invariants for real quadratic number fields.
method Using modified étale cohomology groups and fundamental groups, explicit formulas are derived for real quadratic fields.
result Explicit formulas for mod 2 arithmetic Dijkgraaf-Witten invariants for real quadratic fields are provided.
New invariant for links in 3-sphere computed and computed using diagrams.
problem Computing invariants of links in 3-sphere.
method Defining and computing the parabolic Dijkgraaf-Witten invariant.
result Computed invariants of several links and partial information using diagrams.
New invariant computed for Brieskorn homology spheres.
problem Computing an invariant for specific manifolds.
method Using Dijkgraaf--Witten invariants in topological K-theory.
result Computed invariant for Brieskorn homology spheres.
Generalizes DW invariants for cusped 3-manifolds, distinguishing some pairs.
problem Distinguishing cusped 3-manifolds with same volumes and invariants.
method Introducing and analyzing generalized Dijkgraaf-Witten invariants.
result Generalized DW invariants can distinguish some pairs of cusped hyperbolic 3-manifolds.
Photography method solves manifold invariants.
problem Constructing invariants of manifolds.
method General principle of photography, dealing with various data and data transmission.
result Dijkgraaf-Witten invariants are a specific application of the photography principle.
We introduce defects, with internal gauge symmetries, on a knot and Seifert surface to a knot into the combinatorial construction of finite gauge-group Dijkgraaf-Witten theory. The appropriate initial data for the construction are certain three object categories, with coefficients satisfying a partially degenerate cocy…
Simplified 3D Dijkgraaf-Witten theory with defects explained geometrically.
problem Constructing 3D Dijkgraaf-Witten theory with defects.
method Symmetric monoidal functor from defect cobordism category to vector spaces, using geometric and homotopy theoretic methods.
result Explicit construction of 3D untwisted Dijkgraaf-Witten theory with defects.
New invariant for spin 3-manifolds using super 3-cocycles.
problem Constructing an invariant for spin 3-manifolds.
method State sum construction based on super 3-cocycles and combinatorial representation.
result The invariant is sensitive to the spin structure.
We propose a simple method to produce quandle cocycles from group cocycles, as a modification of Inoue-Kabaya chain map. We further show that, in respect to "universal central extended quandles", the chain map induces an isomorphism between their third homologies. For example, all Mochizuki's quandle 3-cocycles are sho…
T. Mochizuki determined all 3-cocycles of the third quandle cohomologies of Alexander quandles on finite fields. We show that all the 3-cocycles, except those of 2-cocycle forms, are derived from group 3-cocycles of a meta-abelian group. Further, the quandle cocycle invariant of a link using Mochizuki's 3-cocycle is eq…
The paper presents a chain complex for 3-manifold covers, including surface bundles and surgeries.
problem Calculating linking forms and Dijkgraaf-Witten invariants for specific 3-manifold covers.
method Presentation of cellular chain complexes for universal covers of 3-manifolds in a specified class.
result Established a formula for linking forms and developed procedures for Dijkgraaf-Witten invariants.
Summarizes quantum field theories with discrete symmetry, classifying representations and anomalies.
problem Classifying representations and anomalies in quantum field theories with discrete symmetry.
method Classification of representations and anomalies using the ring of profinite integers.
result Rich and complex classification of representations and anomalies.
Discovering topological quantum field theories in 2+1 and 3+1 dimensions.
problem Exploring topological orders in condensed matter lattice models.
method Calculating braiding statistics and link invariants of anyon excitations.
result Identifying new spin topological quantum field theories with specific knot/link invariants.
Develops a new framework for anomaly description in quantum field theories.
problem Anomalies in quantum field theories.
method Extended functorial field theories and symmetric monoidal bicategories.
result Explicit construction of anomaly 2-cocycles and 't Hooft anomalies.
Paper computes motion groups of links using TQFTs, proving a conjecture.
problem Computing representations of motion groups of links in S3. method Dimension reduction of Dijkgraaf-Witten (DW) TQFTs.
result Proves a conjecture for motion groups of closed manifolds and torus links.
This paper approximates SU(2) Chern-Simons theory using finite group gauge theories.
problem Approximating SU(2) Chern-Simons theory with finite group gauge theories.
method Comparing Witten-Reshetikhin-Turaev and Dijkgraaf-Witten invariants on closed 3-manifolds.
result The asymptotics of the DW theory recovers the leading asymptotics of the CS theory at large level.
Constructs a new invariant for 4-manifolds and 3+1 TQFTs.
problem Developing a new invariant for 4-manifolds and 3+1 TQFTs.
method Using a G-crossed braided spherical fusion category to construct a state-sum type invariant.
result Introduces a new invariant for 4-manifolds and 3+1 TQFTs.
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.
This paper demonstrates a topological meaning of quandle cocycle invariants of links with respect to finite connected quandles X, from a perspective of homotopy theory: Specifically, for any prime ℓ which does not divide the type of X, the ℓ-torsion of this invariants is equal to a sum of the colouring po…
We extend the construction of the Hennings TQFT for ribbon Hopf algebras to the case of ribbon quasi-Hopf algebras as defined by Drinfeld. Calculations proceed in a similar fashion to the ordinary Hopf algebra case, but also require the handling of the non-trivial coassociator in the triple tensor product of the algebr…
We develop the theory of Chern-Simons bundle 2-gerbes and multiplicative bundle gerbes associated to any principal G-bundle with connection and a class in $H^4(BG, \ZZ)$ for a compact semi-simple Lie group G. The Chern-Simons bundle 2-gerbe realises differential geometrically the Cheeger-Simons invariant. We apply …
We introduce the concept of twisted contact groupoids, as an extension either of contact groupoids or of twisted symplectic ones, and we discuss the integration of twisted Jacobi manifolds by twisted contact groupoids. We also investigate the very close relationships which link homogeneous twisted Poisson manifolds wit…
Differential characters relate to flat connections' moduli space.
problem Defining homology maps for moduli spaces of flat connections.
method Exploiting Chern-Simons differential characters to define homology maps.
result Established a relationship between differential characters and flat connections' moduli space.
New infinite families of twisted torus knots found.
problem Identifying new types of twisted torus knots.
method Finding new infinite families of twisted torus knots with a single negative twist.
result Eight new infinite families of twisted torus knots are discovered.
The study finds hyperbolic twisted torus links for certain twists.
problem Identifying hyperbolic twisted torus links.
method Analyzing (p,q)-torus links with r strands twisted s times. result All hyperbolic twisted torus links are identified for ∣s∣>3. The paper introduces a method to deform submanifolds in Euclidean space using Drinfel'd twists.
problem Constructing noncommutative deformations of submanifolds in Euclidean space.
method Using Drinfel'd twist deformation of differential geometry, the paper proposes a general procedure to construct noncommutative deformations of an embedded submanifold.
result The method allows for consistent projection of connections and can be applied to various submanifolds like cylinders and hyperboloids.
Paper constructs Chern character for higher twists and shows isomorphism between K-theory and cohomology.
problem Mapping higher twisted K-theory to higher twisted cohomology.
method Constructing Chern character for higher twists and showing isomorphism.
result Chern character gives isomorphism between higher twisted K-theory and higher twisted cohomology.
Paper finds a minimal set of Dehn twists to generate twist subgroup of non-orientable surfaces.
problem Finding a minimal set of Dehn twists to generate the twist subgroup of non-orientable surfaces.
method Used Dehn twists and an argument from Hirose [5] to find a minimal generating set.
result Found a generating set with one less element than the lower bound.
First paper constructs geometric models for twisted differential K-theory.
problem Constructing geometric models for twisted differential K-theory.
method Smooth U(1)-gerbes with connection and twisted vector bundles with connection as cocycles.
result The model satisfies the axioms of Kahle and Valentino.
New algebraic structure for distinguishing twisted virtual handlebody-links.
problem Distinguishing twisted virtual handlebody-links.
method Introducing twisted virtual bikeigebras and using them to define invariants.
result New invariants distinguish some twisted virtual handlebody-links.
The paper studies Gluck twists on 2-knots with periodic monodromy.
problem Determining the resulting 2-knot after a Gluck twist on a 2-knot with periodic monodromy.
method Analyzes the Gluck twist on branched twist spins of 2-knots.
result Provides infinitely many pairs of inequivalent branched twist spins with homeomorphic complements.
Construct noncommutative deformations of algebraic submanifolds in R^n.
problem Deforming algebraic submanifolds in noncommutative geometry.
method Using twisted differential geometry and Drinfel'd twists, constructing noncommutative deformations of algebraic submanifolds.
result Explicitly worked out deformations of quadrics in R^3.
The paper studies the twisted Calabi flow on Kähler manifolds.
problem Analyzing the behavior of the twisted Calabi flow on compact Kähler manifolds.
method Establishing convexity, proving short-time existence, and demonstrating stability of the flow.
result The stability of the twisted Calabi flow near twisted constant scalar curvature Kähler metrics.
Paper proves any twisted link can be described as a unique twisted braid.
problem Understanding twisted links and braids.
method Proved using Alexander and Markov Theorems for classical braids and links.
result Any twisted link can be described as the closure of a unique twisted braid.
Study on singular twisted links and virtual braids, extending knot theory concepts.
problem Extending knot theory concepts to singular twisted links and virtual braids.
method Definition and analysis of singular twisted virtual braids and their monoid structure.
result Presentation of monoid and reduced monoid for singular twisted virtual braids.
In this paper, we develop differential twisted K-theory and define a twisted Chern character on twisted K-theory which depends on a choice of connection and curving on the twisting gerbe. We also establish the general Riemann-Roch theorem in twisted K-theory and find some applications in the study of twisted K-theory o…
Proves existence of Kähler-Einstein metrics with certain twisting forms.
problem Existence of Kähler-Einstein metrics under stability conditions.
method Proves existence assuming twisted K-stability condition, allows certain non-negative twisting forms.
result Existence of twisted Kähler-Einstein metrics under specified conditions.