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. Quantum Frobenius map for SL3 skein modules constructed and described.
problem Constructing a quantum Frobenius map for SL3 skein modules. method Using threading polynomials and the Frobenius map of Parshall-Wang for quantum group Oq(SL3). result Described the quantum Frobenius map for SL3 skein modules. Quantum SL2 reveals surprising cancellations.
problem Understanding surprising cancellations in quantum trace maps.
method Representation theory of quantum group Uq(sl2) and dual Hopf algebra SL2q. result Representation theoretic interpretation of miraculous cancellations.
Quantum trace map connects Teichmüller theory and quantum groups.
problem Connecting quantum groups to Teichmüller theory for knots.
method Quantum snakes technology to relate Fock-Goncharov monodromy matrices to quantum SL_n.
result Quantized Fock-Goncharov matrices satisfy quantum SL_n relations.
The paper studies properties of stated SL(n)-skein algebras and their centers.
problem Properties of stated SL(n)-skein algebras and their centers.
method Quantum trace maps and embeddings into quantum tori.
result Finitely generation and PI-degrees of centers of stated SL(n)-skein algebras.
A Hermitian TQFT from non-semisimple quantum sl(2) modules.
problem Constructing a Hermitian TQFT from a non-semisimple category.
method Endowed a non-semisimple category of quantum sl(2) modules with a Hermitian structure and proved the resulting TQFT is Hermitian.
result Projective representations of the mapping class group in indefinite unitary matrices.
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.
Modified invariants from quantum sl(2|1) for 3-manifolds.
problem Quantum sl(2|1) modules have vanishing quantum dimensions, complicating category construction.
method Specialize q to a root of unity, quotient by morphisms with zero modified quantum dimension, show resulting category is finite and semi-simple.
result Obtained relative G-spherical categories from modified quantum dimensions.
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. Study on quantum sl3 invariant for positive links.
problem Characterizing and understanding the quantum sl3 invariant of positive links. method Skein theory of sl3-webs, explicit formulae, diagrammatic quantities, obstructions. result Positive links are fibered if and only if the second coefficient of the polynomial is 1.
New braid group action defined on projective quantum sl(2) modules.
problem Defining a new braid group action on quantum sl(2) modules.
method Action via R-matrix on tensor powers of simple projective modules.
result The action is faithful for the extended representation.
New central elements found in a quantum algebra related to knot theory.
problem Understanding the algebraic structure of SLd-skein algebras. method Threaded polynomials from symmetric functions.
result Extraction of central elements in SLd-skein algebra. A new invariant for links generalizes Alexander polynomial for sl_3.
problem Defining a non-abelian generalization of the Alexander polynomial.
method Using quantum sl3 representations and Laurent polynomials. result Established a direct relation between Δsl3 and the Alexander polynomial. Study centers of quantum tori and skein algebras for even roots of unity.
problem Understanding the center of quantum tori and skein algebras for even roots of unity.
method Analyzing quantum tori and skein algebras, computing PI-degree, and decomposing matrices.
result PI-degrees of quantum tori and skein algebras are the same.
Quantum trace maps for surfaces are shown to be compatible under triangulations.
problem Constructing and understanding quantum trace maps for surfaces.
method Developed quantum mutation maps between subalgebras of quantum torus algebras for different triangulations.
result Quantum trace maps are natural and independent of triangulation choices.
Temperley-Lieb algebras have been generalized to sl(3) web spaces. Since a cubic bipartite planar graph with suitable directions on edges is a web, the quantum sl(3) invariants naturally extend to all cubic bipartite planar graphs. First we completely classify them as a connected sum of primes webs. We also provide a m…
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. This article gives matrix factorizations for the trivalent diagrams and double line appearing in sln quantum link invariant. These matrix factorizations reconstruct Khovanov-Rozansky homology. And we show that the Euler characteristic of the matrix factorization for a double loop equals the quantum dimens…
We show how to define invariants of graphs related to quantum sl(2) when the graph has more then one connected component and components are colored by blocks of representations with zero quantum dimensions.
We construct a categorification of the quantum sl_3 projectors, the sl_3 analog of the Jones-Wenzl projectors, as the stable limit of the complexes assigned to k-twist torus braids (as k goes to infinity) in a suitably shifted version of Morrison and Nieh's geometric formulation of sl_3 link homology (math.GT/0612754).…
We show that Khovanov homology (and its sl(3) variant) can be understood in the context of higher representation theory. Specifically, we show that the combinatorially defined foam constructions of these theories arise as a family of 2-representations of categorified quantum sl(m) via categorical skew Howe duality. Uti…
We generalize the colored Alexander invariant of knots to an invariant of graphs, and we construct a face model for this invariant by using the corresponding 6j-symbol, which comes from the non-integral representations of the quantum group U_q(sl_2). We call it the SL(2, C) quantum 6j-symbol, and show its relation to t…
Quantum cluster algebra constructed from web skein relations on surfaces.
problem Quantization of cluster structures on moduli spaces of SL3 local systems.
method Constructing a quantum cluster algebra inside the skew-field of a skein algebra of unpunctured surfaces.
result Laurent expressions of webs in clusters have positive coefficients.
We construct a cohomology theory for oriented links using singular cobordisms and a special type of 2-dimensional Topological Quantum Field Theory (TQFT), categorifying the quantum sl(2) invariant. In particular, we give a description of the universal dot-free sl(2) foam cohomology for links via a TQFT.
In this paper I define certain interesting 2-functors from the Khovanov-Lauda 2-category which categorifies quantum sl(k), for any k>1, to a 2-category of universal sl(3) foams with corners. For want of a better name I use the term "foamation" to indicate those 2-functors. I conjecture the existence of similar 2-functo…
We build extensions of the arc rings, relate their centers to the cohomology rings of the Springer varieties, and categorify all level two representations of quantum sl(N).
We associate a moduli problem to a colored trivalent graph; such graphs, when planar, appear in the state-sum description of the quantum sl(N) knot polynomial due to Murakami, Ohtsuki, and Yamada. We discuss how the resulting moduli space can be thought of a representation variety. We show that the Euler characteristic…
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…
Geometrically describes hyperbolic structures on link complements using quantum groups.
problem Describing hyperbolic structures on link complements algebraically.
method Uses octahedral decomposition and Kashaev-Reshetikhin's braiding on quantum group Uξ(sl2). result Shows how to interpret geometrically the algebraic gluing equations for hyperbolic structures.
Unified theories for colored sl(2) knot homology.
problem Equivalence of different models for colored sl(2) knot homology.
method Conceptualized properties into a Chebyshev system and proved its uniqueness.
result Equivalence of Khovanov and Cooper-Krushkal models for the unknot.
Extends quantum trace map to SL3(C) for 3D surfaces.
problem Generalizing quantum trace map to higher dimensions.
method Definition of SL3(C) quantum trace invariant.
result Construction of SL3(C) quantum trace map.
Diagrammatic calculus proves Alexander polynomial formulas.
problem Proving formulas for Alexander polynomial.
method Diagrammatic calculus of quantum sl2 representations. result Quantum invariant determines Alexander polynomial.
New proof of SL(n) skein algebra for twice punctured sphere, showing it's a polynomial algebra.
problem Proving the structure of SL(n) skein algebra for a specific surface.
method Constructing a linear basis of explicit SL(n) webs, proving spanning and linear independence.
result SL(n) skein algebra of twice punctured sphere is a commutative polynomial algebra in n-1 generators.
M. Khovanov and L. Rozansky gave a categorification of the HOMFLY-PT polynomial. This study is a generalization of the Khovanov-Rozansky homology. We define a homology associated to the quantum (sln,∧Vn) link invariant, where ∧Vn is the set of the fundamental representations of the quantum group of $sl…
Study Wilson lines junctions in quantum groups with one-parameter deformations.
problem Understanding local relations of Wilson lines in quantum groups.
method Analyzing junctions of Wilson lines in refined SU(N) Chern-Simons theory and proposing local relations.
result Realization of one-parameter deformations of quantum groups.
Develops Hamiltonian quantization for complex Chern-Simons theory at even level k.
problem Quantum holonomies and representation theory in complex Chern-Simons theory.
method Combinatorial quantization and operator algebra construction.
result Physical Hilbert space identified and Fenchel-Nielsen representation demonstrated.
Tangle functors link quantum representations to knot invariants.
problem Constructing tangle functors from semicyclic representations.
method Developed a tangle functor for framed homogeneous tangles colored with semicyclic representations.
result Invariant of (1,1)-tangles from knots matches Kashaev's invariant. We consider two different quantizations of the character variety consisting of all representations of surface groups in SL_2. One is the skein algebra considered by Przytycki-Sikora and Turaev. The other is the quantum Teichmuller space introduced by Chekhov-Fock and Kashaev. We construct a homomorphism from the skein …
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.
The abstract discusses new 3-manifold invariants and ETQFTs from Lie superalgebra representations.
problem Developing new 3-manifold invariants and ETQFTs from Lie superalgebra representations.
method Examining two m-traces in the category of representations over quantum sl(m∣n), considering quotients, and conjecturing generalizations. result Quotients of perturbative modules over quantum sl(m∣n) lead to 3-manifold invariants and ETQFTs. In this paper, we study the quantum sl(n) representation category using the web space. Specially, we extend sl(n) web space for n≥4 as generalized Temperley-Lieb algebras. As an application of our study, we find that the HOMFLY polynomial Pn(q) specialized to a one variable polynomial …
We define and study the category of symmetric sl2-webs. This category is a combinatorial description of the category of all finite dimensional quantum sl2-modules. Explicitly, we show that (the additive closure of) the symmetric sl2-spider is (braided monoidally) equivalent to …
Extends knot invariant computation to symmetrically colored sl_N.
problem Computing quantum knot invariants for slN. method Develops symmetrically colored R matrix for slN. result Defines FKslN,sym for positive braid knots. This paper quantizes geometry using SL(2,C) Chern-Simons theory and flat connections.
problem Quantizing four-dimensional quantum geometry.
method Using a correspondence between flat connections and four-dimensional simplices, the paper quantizes geometry via complex SL(2,C) Chern-Simons theory.
result The quantum geometrical states are represented by the 3d blocks of analytically continued Chern-Simons theory, and in the semiclassical limit, the three-dimensional Chern-Simons action becomes the discrete Einstein-Hilbert action of a 4-simplex.
Quantum theory constructs a group and skein module for knot complements.
problem Understanding the fundamental group of knot complements using quantum methods.
method Using bottom tangles, the universal space of quantum representations is constructed, then factored by the skein relation to get the skein module.
result Derives recurrence relation for the colored Jones polynomial, known as Aq polynomial. The paper establishes isomorphisms and constructs colored versions of Lawrence representations.
problem Understanding isomorphisms and colored versions of Lawrence representations.
method Explicit isomorphisms and construction of colored versions.
result Matrices for colored versions of BKL and Lawrence representations provided.
New algebraic method for computing TQFT vector spaces.
problem Computing vector spaces for TQFTs from non semi-simple categories.
method Algebraic approach to compute TQFT vector spaces.
result New algebraic method for computing TQFT vector spaces.
The paper proves a quantum modularity conjecture for 3-manifolds.
problem Quantum invariants of 3-manifolds at roots of unity.
method Formulates and proves a strong version of the conjecture for geometric 3-manifolds.
result The conjecture holds for Brieskorn homology spheres and some other examples.