New quantum invariants derived from unrolled quantum groups match existing Hennings invariants.
problem Constructing non-semisimple quantum invariants for 3-manifolds.
method Using unrolled quantum groups at odd roots of unity and small quantum groups.
result Renormalized Hennings invariants coincide with new quantum invariants.
New algebraic setup defines quantum link invariants.
problem Defining and controlling quantum link invariants.
method Quantum Schur--Weyl duality and variants.
result Global definitions of quantum polynomials.
Study Vassiliev invariants for virtual knots, expanding quantum theory.
problem Understanding Vassiliev invariants for virtual knots.
method Define chord diagrams, weight systems, and Lie algebra weight systems for rotational virtual knots.
result Extended quantum invariants capture more information than standard invariants.
Researchers develop a new quantum invariant using a matrix dilogarithm for 3-manifolds.
problem Developing quantum invariants for 3-manifolds.
method Using a sl3 matrix dilogarithm and quantum groups. result The sl3 matrix dilogarithm can be considered as a 6j-symbol. We consider quantum invariants of 3-manifolds associated with arbitrary simple Lie algebras. Using the symmetry principle we show how to decompose the quantum invariant as the product of two invariants, one of them is the invariant corresponding to the projective group. We then show that the projective quantum invarian…
Survey of quantum enhancements in knot theory.
problem Classifying knots using quantum invariants.
method Collecting and analyzing various quantum invariants.
result New invariants defined by coloring knots with algebraic objects.
In this paper we give a re-normalization of the Reshetikhin-Turaev quantum invariants of links, by modified quantum dimensions. In the case of simple Lie algebras these modified quantum dimensions are proportional to the usual quantum dimensions. More interestingly we will give two examples where the usual quantum dime…
Introduces geometric quantization and Witten's quantum invariants.
problem None explicitly stated; focuses on introduction.
method Expository introduction to geometric quantization and Witten's quantum invariants.
result Introduction to geometric quantization and Witten's quantum invariants.
Quantum Teichmüller invariants studied for 3-manifolds.
problem Quantum invariants of 3-manifolds.
method Characterization through intertwiners and topological quantum field theories.
result Invariants of mapping tori defined by traces of intertwiners.
Enhances quantum invariants using tribracket brackets.
problem Quantum invariants of tribracket-colored knots and links.
method Introduces tribracket brackets as skein invariants.
result Provides new quantum invariants and examples.
Quantum link invariants derived from skein algebras.
problem Defining invariants for framed links with SL2 local systems.
method Theory of representations of stated skein algebras, quantum coadjoint action, Drinfeld double, Bonahon-Wong quantum trace.
result Explicit formulas for link invariants and alternative proof of Murakami-Murakami relation.
Quantum invariants of 3-manifolds and links reviewed, with connections to other topological invariants.
problem Quantum invariants of 3-manifolds and links.
method Review of recent developments and connections to other invariants.
result Rich features of quantum invariants like quantum modularity and Verma module structures.
Quantum invariants from Uhsl(2∣1) are q-holonomic.
problem Understanding quantum invariants from a specific quantum group.
method Demonstrated q-holonomic property through quantum group representations.
result Existence of an underlying field theory for these quantum invariants.
A new quantum gauge model is proposed. From this quantum gauge model we derive a quantum invariant of 3-manifolds. We show that this quantum invariant of 3-manifolds gives a classification of closed (orientable and connected) 3-manifolds. From this classification we then prove the Poincaré conjecture.
This paper defines the concept of an oriented quantum algebra and develops its application to the construction of quantum link invariants. We show that all known quantum link invariants can be put into this framework.
State sums for quantum link invariants from a specific representation.
problem Calculating quantum link invariants from a specific representation of U_q(gl_{N|M}).
method Using state sums and representation theory of U_q(gl_{N|M}).
result Explicit relation with Kashaev invariants for the N-th exterior power of the standard representation.
Quantum invariant derived from ternary cohomology of self-distributive structures.
problem Defining and proving a quantum invariant from ternary cohomology.
method Constructing a ribbon category from a TSD set, showing it coincides with the cocycle invariant.
result The ribbon cocycle invariant is a quantum invariant.
Researchers prove quantum invariants remain hard even when restricted.
problem Computing quantum invariants on 3-manifolds with specific restrictions.
method Using Heegaard splittings and Hempel distance, they construct a hyperbolic 3-manifold with same invariant.
result Proving hardness of computing quantum invariants is preserved under specific restrictions.
In this thesis, we give a unification of the quantum WRT invariants. Given a rational homology 3-sphere M and a link L inside, we define the unified invariants, such that the evaluation of these invariants at a root of unity equals the corresponding quantum WRT invariant. In the SU(2) case, we assume the order of the f…
Quantum cocycle invariants derived from Yang-Baxter cohomology.
problem Constructing stronger quantum knot invariants.
method Developing quantum cocycle invariants using Yang-Baxter cohomology and deformation theory.
result Quantum cocycle invariants yield stronger invariants in certain examples.
Study uses big data to analyze quantum invariants.
problem Investigate structural properties of Jones polynomial.
method Exploratory and topological data analysis, including coloring, rank increase, categorification.
result Contrasts behavior of Jones polynomial under various enhancements.
Quantum group invariants from Lie superalgebra representations at roots of unity.
problem Constructing invariants for links and 3-manifolds from quantum group representations.
method Using nilpotent irreducible representations of quantum group Uξsl(2∣1), modified trace, and relative G-modular category. result Link and 3-manifold invariants constructed from quantum group Uξsl(2∣1) representations at roots of unity. Quantum invariants help distinguish knots via algebra and equations.
problem Determine if two knots are equivalent.
method Use Hopf algebras and solutions to the Yang-Baxter equation.
result Quantum invariants are powerful tools for knot classification.
Paper defines a polynomial invariant for surface-links using quantum A_2 invariant.
problem Defining a polynomial invariant for surface-links.
method Using the quantum A_2 invariant and Yoshikawa moves, a polynomial is defined for marked graph diagrams.
result The polynomial invariant is useful for studying ribbon 2-knots.
We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl(2) and sl(3) and by Mazorchuk-Stroppel and Sussan for sl(n). Our technique uses categorifications of the tensor produ…
Modified Hennings invariant defined using quantum groups and integrals.
problem Defining a modified Hennings invariant using quantum groups.
method Topological ribbon Hopf algebra, discrete Fourier transforms, symmetrized graded integral, modified trace.
result Modified graded Hennings invariant defined and extended to empty manifolds.
For each finite dimensional, simple, complex Lie algebra g and each root of unity ξ (with some mild restriction on the order) one can define the Witten-Reshetikhin-Turaev (WRT) quantum invariant τMg(ξ)∈C of oriented 3-manifolds M. In the present paper we construct an invariant…
For every rational homology 3-sphere with 2-torsion only we construct a unified invariant (which takes values in a certain cyclotomic completion of a polynomial ring), such that the evaluation of this invariant at any odd root of unity provides the SO(3) Witten-Reshetikhin-Turaev invariant at this root and at any even …
The volume conjecture is extended for surface diffeomorphisms with quantum invariants.
problem Extending the volume conjecture for quantum invariants of surface diffeomorphisms.
method Relating asymptotics of quantum invariants to hyperbolic cone structures on mapping tori.
result The conjecture is proven for a specific case of the once-punctured torus bundle.
New invariants derived from Kauffman bracket for 3-manifolds.
problem Quantum invariants of 3-manifolds, especially non-semisimple ones.
method Combinatorial methods using Temperley-Lieb algebras and Kauffman bracket polynomials.
result Recovery of invariants from small quantum group of sl2. Defines a new link invariant for type D webs.
problem Invariants for framed unoriented links in type D.
method Positive state sum for type D webs, derived from quantum so(2N) modules.
result Relates to Reshetikhin-Turaev's invariant.
We study quantum invariant Z(M) for cusped hyperbolic 3-manifold M. We construct this invariant based on oriented ideal triangulation of M by assigning to each tetrahedron the quantum dilogarithm function, which is introduced by Faddeev in studies of the modular double of the quantum group. Following Thurston and Neuma…
The paper constructs quantum invariants for knotoid diagrams.
problem Quantum invariants for knotoid diagrams in R2. method Decompose Morse knotoid diagrams into basic elementary diagrams, each associated with a matrix solving the quantum Yang-Baxter equation. Define quantum state sum models to recover various polynomials.
result Recover and define new polynomials for Morse knotoids.
New knot invariants derived using quantum cluster algebras.
problem Deriving new knot invariants from quantum cluster algebras.
method Interpreting R-matrix of Uq(sl2) as cluster transformation, introducing auxiliary parameter ε. result Derives perturbed-Alexander invariants with higher-order terms in ε. New quantum invariant is asymptotically multiplicative under cyclic covers.
problem Quantum invariants are not multiplicative under finite covers.
method Introduced a perturbative power series invariant of cusped hyperbolic 3-manifolds.
result The power series is asymptotically multiplicative under cyclic covers.
New quantum invariants for 3-manifolds with boundary defined via virtual links.
problem Quantum invariants for 3-manifolds with non-vacuous boundaries.
method Surgery on manifolds of the form FimesI and virtual link diagrams. result First meaningful, nontrivial, calculable quantum invariants of 3-manifolds with non-vacuous boundary.
A new quantum invariant for virtual knots and links.
problem Quantum invariants for virtual knots and links.
method Generalization of biquandle brackets to parity biquandles.
result The new invariant is stronger than classical biquandle brackets for virtual knots.
We study quantum moment maps of G-invariant star products, which are a quantum analogue of the moment map for classical Hamiltonian systems. Introducing an integral representation, we show that any quantum moment map for a G-invariant star product is differentiable. This property gives us a new method for the class…
Quantum invariants derived from Uq(sl2) link holonomy.
problem Quantum invariants of links and their relations.
method Using quantum groups and Schur-Weyl duality, constructing quantum holonomy invariants.
result Quantum invariants of links can be derived from Uq(sl2) representations. In GT/0006019 oriented quantum algebras were motivated and introduced in a natural categorical setting. Invariants of knots and links can be computed from oriented quantum algebras, and this includes the Reshetikhin-Turaev theory for Ribbon Hopf algebras. Here we continue the study of oriented quantum algebras from a m…
New invariants for 3-manifolds from foliations and noncommutative geometry.
problem Creating new invariants for 3-manifolds.
method Using taut codim-1 foliations and noncommutative geometry techniques.
result Definition of new invariants for 3-manifolds.
An earlier article with Francis Bonahon introduced new invariants for pseudo-Anosov diffeomorphisms of surface, based on the representation theory of the quantum Teichmuller space. We explicity compute these quantum hyperbolic invariants in the case of the 1-puncture torus and the 4-puncture sphere.
We introduce a new class of quantum enhancements we call biquandle brackets, which are customized skein invariants for biquandle colored links.Quantum enhancements of biquandle counting invariants form a class of knot and link invariants that includes biquandle cocycle invariants and skein invariants such as the HOMFLY…
We construct knot invariants from the radical part of projective modules of restricted quantum groups. We also show a relation between these invariants and the colored Alexander invariants.
Logarithmic invariant for restricted quantum sl(2) constructed.
problem Constructing a logarithmic invariant for a specific quantum group.
method Combining a universal invariant and a modified trace, defined for a 3-manifold and link.
result A new logarithmic invariant for restricted quantum sl(2) at a 2p-th root of unity.
Modified knotoids with framing and coframing for quantum invariants.
problem Defining and classifying knotoids with framing.
method Defining framed and biframed knotoids, showing topological correspondence, and constructing quantum invariants.
result Generalized quantum knotoid invariants constructed.
Infinite families of quantum modular invariants for 3-manifolds are discovered.
problem Quantum modular forms and invariants of 3-manifolds.
method Lattice cohomology and BPS q-series of 3-manifolds, recently developed theory by AJK. result First calculation of AJK series for an infinite family of 3-manifolds.
This paper studies rotational virtual knot theory and its relationship with quantum link invariants. Every quantum link invariant for classical knots and links extends to an invariant of rotational virtual knots and links. The paper sets up the background virtual knot theory, defines rotational virtual knot theory, stu…