In this work, differential geometry of the Z-graded quantum superplane is constructed. The corresponding quantum Lie superalgebra and its Hopf algebra structure are obtained.
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Differential geometry of the quantum Lie superalgebra of the extended quantum superplane and its Z-graded Hopf algebra structure is obtained. Its Z-graded dual Hopf algebra is also given.
In this work, the Z-graded differential geometry of the quantum plane is constructed. The corresponding quantum Lie algebra and its Hopf algebra structure are obtained. The dual algebra, i.e. universal enveloping algebra of the quantum plane is explicitly constructed and an isomorphism between the quantum Lie algeb…
Extreme Khovanov spectramath.GT
New proof of Khovanov spectrum equivalence at extreme grading.
problem Proving homotopy equivalence of spectra at extreme quantum grading.
method Stable homotopy equivalence proof using González-Meneses et al. spectrum and Lipshitz-Sarkar Khovanov spectrum.
result Stable homotopy equivalence between the two spectra at extreme quantum grading.
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.
Unified model for knot polynomials using quantum Heegaard diagrams.
problem Categorify knot polynomials using Floer homology.
method Construct quantum Heegaard diagrams, identify gradings, and define a two-variable graded intersection.
result Unified intersection model recovers Alexander and Jones polynomials.
3D HQFTs constructed using graded monoidal categories.
problem Constructing 3D HQFTs with specific targets.
method Using spherical χ-fusion categories and the state sum method.
result 3D HQFTs constructed with target Bχ.
Develops non-semisimple ETQFTs for 3-manifolds.
problem Creating ETQFTs for non-semisimple categories.
method Introducing relative modular categories and using 2-categorical universal construction.
result Extends ETQFTs to non-semisimple cases.
Defines a new 2+1-G-HQFT using graded skein modules.
problem Developing a new quantum field theory for groups.
method G-graded chromatic maps and skein modules.
result Recover modified Turaev-Viro invariants.
-quantum invariants from an intersection of two Lagrangians in a symmetric power of a surfacemath.GT
The paper connects quantum invariants to intersections of Lagrangians in symmetric power spaces.
problem Computing colored Jones and Alexander polynomials.
method Using two Lagrangians in a symmetric power of a surface to compute polynomials.
result Colored Jones and Alexander polynomials are special cases of a graded intersection between Lagrangians.
Short note observes quantum Hochschild homology as a composition of known operations.
problem Quantum Hochschild homology as a new invariant of annular links.
method Observes quantum Hochschild homology as a composition of two known operations.
result Quantum Hochschild homology is a valid invariant of annular links.
Given a unital associatve graded algebra we construct the graded q-differential algebra by means of a graded q-commutator, where q is a primitive N-th root of unity. The N-th power (N>1) of the differential of this graded q-differential algebra is equal to zero. We use our approach to construct the graded q-differentia…
We generalize categorified Jones-Wenzl projectors in odd Khovanov homology.
problem Categorify Jones-Wenzl projectors for odd Khovanov homology.
method Develop grading multicategories to replace grading categories, proving existence and uniqueness of categorified projectors.
result Existence and uniqueness of categorified Jones-Wenzl projectors in odd Khovanov homology.
The paper connects knot homology, quantum 6j-symbols, and complements of knots.
problem Investigating the relationship between knot homology, quantum 6j-symbols, and knot complements.
method Developed a grading rule for HOMFLY-PT and Kauffman homology, found relationships between A-polynomials, and conjectured closed-form expressions for quantum 6j-symbols and knot complements.
result Closed-form expressions for SO(N) quantum 6j-symbols and conjectured expressions for (a,t)-deformed F_K for knot complements.
In earlier work of the authors, the Khovanov complex of a knot or link appeared as the first page in a spectral sequence abutting to the instanton homology. The quantum and (co)homological gradings on Khovanov homology do not survive as gradings, but we show that they survive as filtrations.
Quantum homology theory for links in annuli using traces and quantum groups.
problem Developing a new triply graded link homology theory.
method Quantizing a general theory of traces and shadows for a bicategory, using and -deformed structures.
result Quantum annular link homology theory with an action of and braid group.
Develops higher representation theory for odd Khovanov homology and rewriting theory.
problem Quantum topology and higher algebraic structures.
method Higher representation theory and rewriting theory applied to Khovanov homology.
result Established a basis theorem for graded -foams.
Quantum method categorifies knot polynomial.
problem Alexander polynomial of knots.
method Modified foam evaluation, algebraic approach.
result Quantum categorification of Alexander polynomial.
Topological model for coloured Alexander invariants from quantum group representations.
problem Quantum invariants from at roots of unity.
method Topological model using graded intersection pairings in a covering space.
result Coloured Alexander invariants can be obtained as homology class pairings.
The Jones-Wenzl projectors play a central role in quantum topology, underlying the construction of SU(2) topological quantum field theories and quantum spin networks. We construct chain complexes whose graded Euler characteristic is the "classical" projector in the Temperley-Lieb algebra. We show that they are homotopy…
Extends gl(m|k) construction using Hilbert scheme of points.
problem Quantum invariants of super-algebras gl(m|k).
method Using Hilbert scheme of points on C^2.
result Constructs triply graded link homology for gl(m|k).
The article explores causal structures in symmetric spaces and their relation to AQFT.
problem Understanding causal structures in symmetric spaces and their applications in AQFT.
method Classification of reductive causal symmetric spaces using Euler elements and 3-grading.
result Extraction of real Matsuki crowns and description of stabilizer groups of Euler elements.
Rank inequality proven for annular Khovanov homology of 2-periodic links.
problem Proving a rank inequality for annular Khovanov homology of 2-periodic links.
method Spectral sequence analysis with quantum and sl2 weight spaces.
result Proven rank inequality .
Exposes graded and microformal geometry, focusing on -manifolds.
problem Describes new geometric structures and their applications.
method Introduces -manifolds and microformal geometry.
result Establishes connections between -manifolds and Lie algebras.
New knot homology invariant grows exponentially with color.
problem Constructing and understanding colored torus knot homology.
method Invariant construction and recursive formula for reduced HOMFLY homology.
result Doubly-graded invariant of positive torus knots grows exponentially in color.
Reproduces basic supersymmetric QFTs using complexified graded algebraic geometry.
problem Understanding and describing supersymmetric quantum field theories.
method Tower construction in complexified Z/2-graded C-infinity-algebraic geometry and a purge-evaluation/index-contracting map.
result Reproduces , Wess-Zumino model and gauge theory.
The paper introduces transposed Poisson superalgebras and their properties.
problem Developing algebraic structures for quantum physics.
method Defining transposed Poisson superalgebras using derivations and proving their identities.
result Six identities are crucial for understanding transposed Poisson superalgebras.
SQS uses quantum kernels to improve credit scoring with fewer data points.
problem Credit scoring models struggle with scarce and skewed data.
method Systemic Quantum Score (SQS) leverages quantum kernels for better pattern extraction.
result SQS shows improved performance and pattern extraction with fewer data points.
Quantum torus methods enhance understanding of skein algebras and modules.
problem Investigating Kauffman bracket skein algebras and modules using quantum torus methods.
method Two quantum torus methods: embedding into quantum Teichmüller space and filtering to a monomial subalgebra.
result Generalized Chebyshev homomorphism and refined unicity theorem for skein modules.
We show that the small quantum product of the generalized flag manifold is a product operation on $H^*(G/B)\otimes \bR[q_1,..., q_l]$ uniquely determined by the fact that it is a deformation of the cup product on , it is commutative, associative, graded with respect to , it satisfies a certain…
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.
New formula calculates knot polynomials for rectangular representations.
problem Calculating knot polynomials for arbitrary rectangular representations.
method Rewrote differential expansion formula for HOMFLY polynomials, using quantum dimensions of symmetric representations.
result Rectangular superpolynomials are positive Laurent polynomials.
The paper studies Turaev genus one links and finds their Khovanov homology to be isomorphic to Z in at least one extremal grading.
problem Understanding the extremal properties of Turaev genus one links.
method Analyzing Khovanov homology of Turaev genus one links.
result Khovanov homology of Turaev genus one links is isomorphic to Z in at least one extremal grading.
A new model calculates colored Jones polynomials using homology.
problem Calculating colored Jones polynomials for different colors.
method Using a homological model based on the Lawrence representation and Kohno's result.
result Colored Jones polynomials are described as a graded intersection pairing of homology classes.
New model for Witten-Reshetikhin-Turaev invariants using Lagrangian intersections.
problem Categorification of Witten-Reshetikhin-Turaev invariants for 3-manifolds.
method Constructing a topological model from quantum group U_q(sl(2)) using Lagrangian intersections in configuration spaces.
result Witten-Reshetikhin-Turaev invariants are encoded by intersections of Lagrangian submanifolds in a fixed configuration space.
The paper sets genus bounds for twisted quantum invariants.
problem Bounding the degree of twisted quantum invariants for knots.
method Using Reshetikhin-Turaev construction and Drinfeld doubles.
result Degree of polynomials is bounded by .
Quite a number of -gradings, , appear in Physics and in Mathematics. The corresponding sign rules are given by the `scalar product' of the involved -degrees. The new theory exhibits challenging differences with the classical one: nonzero degree even coordinates are not nilpotent…
The paper introduces a quantum state system to count perfect matchings in graphs.
problem Counting perfect matchings in graphs using quantum state systems.
method Topological quantum field theory (TQFT) and spectral sequences.
result The filtered -color vertex homology for is generated by perfect matchings.
New invariant from quantum algebra for 3-manifold bundles.
problem Quantum invariants of flat 2-bundles over 3-manifolds.
method From an involutory Hopf algebra graded by a crossed module, constructing a homotopy invariant via -colored Heegaard diagrams.
result Reduces to Kuperberg invariant when bundle is trivializable.
This paper is an introduction to Khovanov homology, starting with the Kauffman bracket state summation, emphasizing the Bar-Natan Canopoloy and tangle cobordism approach. The paper discusses a simplicial approach to Khovanov homology and a quantum model for it so that the graded Euler characteristic that produces the J…
Quantum invariants are explained as intersections in configuration spaces.
problem Quantum invariants of knots and links.
method Topological intersections in configuration spaces.
result Coloured Jones and Alexander polynomials are special cases of intersection pairings.
What is a Singular Knot?math.GT
Study of singular knots connects knot theory with quantum algebra.
problem Understanding the structure of knots with transverse double points.
method Analyzes singular knots and their relationship to Vassiliev invariants and quantum algebra.
result Extensions of non-numerical knot invariants to singular knots have been explored.
The paper connects knot Floer homology to quantum representations.
problem Decategorification of knot Floer homology using bordered theory.
method Relating decategorifications to representations of U_q(gl(1|1)).
result Identifies decategorifications with tensor products of representations.
The general method of Reshetikhin and Turaev is followed to develop topological invariants of closed, connected, orientable 3-manifolds from a new class of algebras called pseudo-modular Hopf algebras. Pseudo-modular Hopf algebras are a class of Z_2-graded ribbon Hopf algebras that generalise the concept of a modular H…
Floer theory constructs filtrations on quantum cohomology for symplectic manifolds.
problem Quantum cohomology of symplectic manifolds with -actions.
method Floer theory applied to -actions on symplectic manifolds.
result Constructs a family of filtrations on quantum cohomology for Conical Symplectic Resolutions.
Researchers compute determinants and torsions of Rumin complex in specific Lie group representations.
problem Computing determinants and torsions of Rumin complex in specific Lie group representations.
method Analyzing Schrodinger and generic representations of the (2,3,5) nilpotent Lie group.
result Computed the spectrum and zeta regularized determinant of Rumin differentials in Schrodinger representations and evaluated their alternating product in generic representations.
Functor decomposes Khovanov spectra for non-alternating diagrams.
problem Computing Khovanov spectra for diagrams without alternating pairs.
method Functor from cube to Burnside 2-category, decomposition into simplicial complexes.
result Homotopy type of almost-extreme Khovanov spectra computed.
Heegaard Floer theory is a kind of topological quantum field theory, assigning graded groups to closed, connected, oriented 3-manifolds and group homomorphisms to smooth, oriented 4-dimensional cobordisms. Bordered Heegaard Floer homology is an extension of Heegaard Floer homology to 3-manifolds with boundary, with ext…