Study Legendrian graph invariants via augmentation and ruling polynomials.
problem Equivalence of Legendrian isotopy invariants.
method Use augmentation number and ruling polynomial for front projection.
result Show equivalence between augmentation number and ruling polynomial.
Legendrian contact homology studies knots in 3D space.
problem Understanding knots in 3D space.
method Legendrian contact homology and differential graded algebra.
result New insights into knot behavior in R3. New algebra defined for Legendrian submanifolds, preserving key invariants.
problem Defining a new algebra to preserve invariants of Legendrian submanifolds.
method Combining string topology techniques with combinatorial methods to count holomorphic disks.
result The new algebra PDA is a filtered, differential graded algebra that captures invariants of Legendrian submanifolds. A new algebra for Legendrian graphs defined combinatorially.
problem Defining an algebra for Legendrian graphs and tangles.
method Combinatorial approach using Legendrian contact homology.
result Shows a van Kampen type theorem for differential graded algebras.
The study extends Legendrian link rulings and augmentations in a specific geometric space.
problem Defining and characterizing augmentations and rulings for Legendrian links in a complex geometric setting.
method Extending the definition of normal rulings and using Chekanov-Eliashberg differential graded algebras to establish equivalences.
result Existence of augmentations is equivalent to the existence of normal rulings, with implications for symplectic homology.
The Chekanov-Eliashberg differential graded algebra of a Legendrian knot L is a rich source of Legendrian knot invariants, as is the theory of generating families. The set P(L) of homology groups of augmentations of the Chekanov-Eliashberg algebra is an invariant, as is a count of objects from the theory of generating …
Given a front projection of a Legendrian knot K in R3 which has been cut into several pieces along vertical lines, we assign a differential graded algebra to each piece and prove a van Kampen theorem describing the Chekanov-Eliashberg invariant of K as a pushout of these algebras. We then use this the…
For any Legendrian knot in (R^3,ker(dz-ydx)), we show that the existence of an augmentation to any field of the Chekanov-Eliashberg differential graded algebra over Z[t,t^{-1}] is equivalent to the existence of a ruling of the front diagram, generalizing results of Fuchs, Ishkhanov, and Sabloff. We also show that any e…
We establish tools to facilitate the computation and application of the Chekanov-Eliashberg differential graded algebra (DGA), a Legendrian-isotopy invariant of Legendrian knots in standard contact three-space. More specifically, we reformulate the DGA in terms of front projection, and introduce the characteristic alge…
We study satellites of Legendrian knots in R^3 and their relation to the Chekanov-Eliashberg differential graded algebra of the knot. In particular, we generalize the well-known correspondence between rulings of a Legendrian knot in R^3 and augmentations of its DGA by showing that the DGA has finite-dimensional represe…
Study Legendrian links using representations and sheaves.
problem Understanding Legendrian links through algebraic and geometric representations.
method Investigate an A∞ category of n-dimensional representations and conjecture equivalence to sheaves. result Established cohomological equivalence for Legendrian (2,m) torus links. New algebra invariant distinguishes Legendrian knots in convex surfaces.
problem Distinguishing Legendrian knots in convex surfaces using invariants.
method Defined a differential graded algebra (DGA) for Legendrian knots in thickened convex surfaces, generating it from Reeb chords and counting immersed polygons.
result The stable tame isomorphism type of the DGA is invariant under Legendrian isotopy and can distinguish knots not distinguishable by classical invariants.
New algebra structure for Legendrian knots preserves contact homology invariants.
problem Constructing an L∞ algebra for Legendrian knots. method Combining rational Symplectic Field Theory and combinatorial methods.
result Invariant Poisson algebra of Legendrian links under isotopy.
For a Legendrian knot L in R^3 with a chosen Morse complex sequence (MCS) we construct a differential graded algebra (DGA) whose differential counts "chord paths" in the front projection of L. The definition of the DGA is motivated by considering Morse-theoretic data from generating families. In particular, when the MC…
Study uses barcode theory to bound displacement energy of Legendrian submanifolds.
problem Bounding displacement energy for Legendrian submanifolds.
method Applies barcodes of persistent homology to Chekanov-Eliashberg algebra, linearizing only below a certain action level.
result Shows Legendrians that admit augmentations cannot be C0-approximated by stabilized Legendrians. The paper integrates DGLA to DGLG using HCPs and Hopf algebras.
problem Integrating DGLA to DGLG.
method Definition of DGLG and HCPs, use of graded Hopf algebras.
result Construction of DGLG from DGLA and vice versa.
New formulas link knot invariants from DGA and satellite polynomials.
problem Establishing relationships between knot invariants.
method Introducing new polynomials and formulas linking DGA and satellite invariants.
result Arbitrary m-graded satellite ruling polynomials are determined by DGA of K.
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 link Ginzburg algebras to Weinstein manifolds and Legendrian knots.
problem Understanding the relationship between Ginzburg algebras and Weinstein manifolds.
method Associated a stopped Weinstein manifold to a quiver and subquiver, proving quasi-isomorphism of relative Ginzburg algebra and Chekanov-Eliashberg dg-algebra.
result Relative Ginzburg algebra is quasi-isomorphic to Chekanov-Eliashberg dg-algebra of a singular Legendrian unknot link.
This is the second in a series of papers laying the foundations for a differential graded approach to derived differential geometry (and other geometries in characteristic zero). In this paper, we extend the classical notion of a dg-algebra to define, in particular, the notion of a differential graded algebra in the wo…
The paper examines smoothness in graded skew Clifford algebras.
problem Smoothness of graded skew Clifford algebras.
method Investigation of differential smoothness.
result Results on the differential smoothness of graded skew Clifford algebras.
Determines algebra structure of complex differential forms operators.
problem Identifying the algebra structure of differential operators on complex-valued differential forms.
method Shows it is the universal enveloping algebra of a graded Lie algebra and determines its cohomology.
result Determines the cohomology of the graded Lie algebra with respect to various inner differentials.
Expands on graded Poisson algebras, their properties, and applications.
problem None explicitly stated; focuses on overview and properties.
method Overview and discussion of properties and applications.
result Provides detailed overview of graded Poisson algebras and their contexts.
We study the unwrapped Fukaya category of Lagrangian branes ending on a Legendrian knot. Our knots live at contact infinity in the cotangent bundle of a surface, the Fukaya category of which is equivalent to the category of constructible sheaves on the surface itself. Consequently, our category can be described as cons…
New algebra pong algebra computed for knot Floer homology.
problem Computing A-infinity structure on knot Floer homology.
method Introduced differential graded algebra, pong algebra.
result Computed A-infinity structure on pong algebra's homology.
This paper develops a theory of graded manifolds in differential geometry.
problem Defining consistent global descriptions of graded manifolds with mixed graded coordinates.
method Using sheaves of graded commutative associative algebras on topological spaces.
result Resolved known issues in the definition of graded manifolds, especially those involving mixed graded coordinates.
We introduce the concept of N-differential graded algebras (N-dga), and study the moduli space of deformations of the differential of a N-dga. We prove that it is controlled by what we call the N-Maurer-Cartan equation.
We construct families of differential graded algebras R and R \boxtimes R and give an algebraic formulation of the contact category of a disk through the differential graded category DGP(R) generated by some distinguished projective differential graded R-modules. The homology category H^0(DGP(R)) is a triangulated cate…
Constructs a new graded variety from algebraic data.
problem Creating a Z-graded extension of differential varieties. method Algorithm using homotopy retract data of Koszul-Tate resolution.
result Significantly reduced number of homological computations.
In this work, the Z3-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…
New calculus framework for vector bundles with metrics.
problem Developing calculus for vector bundles with fiber metrics.
method Adapting differential calculus to graded commutative algebras and focusing on diole and triole algebras.
result Triole algebra provides a suitable environment for vector bundle calculus with fiber metrics.
Differential geometry of the quantum Lie superalgebra of the extended quantum superplane and its Z2-graded Hopf algebra structure is obtained. Its Z2-graded dual Hopf algebra is also given.
The paper introduces Z-graded manifolds and their properties.
problem Defining functions and vector fields on Z-graded manifolds. method Introduced Z-graded manifolds from local models, defined functions and vector fields, and reviewed correspondences. result Properties and definitions of Z-graded manifolds. Generators and relations for knot Floer homology algebras computed.
problem Computing knot Floer homology using algebras defined by Ozsváth and Szabó.
method Generators and relations description, homology computation, formality determination.
result Generators and relations for the algebras are found, and their homology is computed.
The article introduces a combinatorial differential algebra for cubic planar graphs.
problem Understanding the structure of cubic planar graphs.
method Defining a combinatorial differential graded algebra based on binary sequences and counting.
result The algebra's graded augmentation variety's rational points match (q+1)-colorings of the dual graph.
New algebraic formalism for differential calculus in Diolic algebras.
problem Studying differential calculus in vector bundles.
method Introducing functors of differential calculus over arbitrary graded commutative algebras (DCGCA) and applying this to Diolic algebras.
result Recovery of well-known objects and notions from ordinary differential, symplectic, and Poisson geometry, with unique aspects.
A Lie-infinity morphism relates derived Leibniz algebras from Maurer-Cartan elements.
problem Relating derived Leibniz algebras via Lie-infinity morphisms.
method Constructing a Leibniz-infinity morphism from a Lie-infinity morphism and Maurer-Cartan elements.
result Derived Leibniz algebras are related by a Leibniz-infinity morphism.
For Legendrian links in the 1-jet space of S1 we show that the 1-graded ruling polynomial may be recovered from the Kauffman skein module. For such links a generalization of the notion of normal ruling is introduced. We show that the existence of such a generalized normal ruling is equivalent to sharpness of the Kau…
A formula connects two algebraic structures derived from a category.
problem Connecting two algebraic structures derived from a category.
method Using differential graded modular functors and Calabi-Yau structures.
result The action of a specific mapping class group element transforms one algebraic structure into another.
In this work, differential geometry of the Z3-graded quantum superplane is constructed. The corresponding quantum Lie superalgebra and its Hopf algebra structure are obtained.
Generalizes Lie algebra of conformal Killing vector fields to conformal Killing-Yano forms.
problem Understanding the algebraic structure of conformal Killing-Yano forms.
method Proposes a new Lie bracket and shows graded Lie algebra properties in constant and Einstein manifolds.
result Conformal Killing-Yano forms form a graded Lie algebra in specific manifolds.
We prove that the isomorphism type of a large class of groups (containing finite groups, countable Artinian groups and mapping class groups of certain surfaces, among others) is determined by the set of differential graded Q-algebras on which these groups act faithfully.
New algebras model Ozsváth-Szabó's Kauffman-states.
problem Modeling Ozsváth-Szabó's Kauffman-states.
method Defined new differential graded algebras A(n,k,S) and showed quasi-isomorphism to B(n,k,S).
result Natural emergence of Ozsváth-Szabó's gradings.
Three new types of graded Lie groups are constructed and analyzed.
problem Generalizing Lie theory to Z-graded geometry. method Direct geometric construction and functor-of-points perspective.
result Isomorphic Lie algebras of the new graded Lie groups.
Constructs mixed Hodge structures on Kähler manifolds.
problem Real variations of mixed Hodge structures over compact Kähler manifolds.
method Using Sullivan's 1-minimal models of differential graded algebras associated with real variations of Hodge structures.
result Constructs real variations of mixed Hodge structures.
Let g be a simplicial Lie algebra with Moore complex Ng of length k. Let G be the simplicial Lie group integrating g, which is simply connected in each simplicial level. We use the 1-jet of the classifying space of G to construct, starting from g, a Lie k-algebra L. The so constructed Lie k-algebra L is actually a diff…
Study vector fields and derivations on differentiable stacks.
problem Understanding structures on differentiable stacks.
method Introduced module structures on dgla of multiplicative vector fields and graded algebra of functions on Lie groupoids.
result Associated structure of a graded Lie-Rinehart algebra on vector fields is Morita invariant.
We construct geometric examples of N-differential graded algebras such as the algebra of differential forms of depth N on an affine manifold, and N-flat covariant derivatives.