arXiv research

Obstruction theory for complex bigraded differential algebras.

The study introduces new foliations and structures on complex manifolds.

New knot invariant from Kauffman states and algebraic modules.

Develops a new method for constructing absolute parallelisms on CR structures.

The paper explains why a specific type of link homology is useful.

Study bigraded formality and Aeppli-Bott-Chern-Massey products on complex manifolds.

We show that the Malcev Lie algebra of the fundamental group of a compact $2n+1$-dimensional Sasakian manifold with $n\ge 2$ admits a quadratic presentation by using Morgan's bigradings of minimal models of mixed-Hodge diagrams. By using bigradings of minimal models, we also simplify the proof of the result of Cappelle…

New CR invariant treatment of Rumin complex via differential forms.

We define a new algebra for double vector bundles, linking it to Lie algebroids.

Nontrivial Massey products found on compact Kähler manifolds.

Let $M= G/Γ$ be a compact nilmanifold endowed with an invariant complex structure. We prove that, on an open set of any connected component of the moduli space ${\cal C} ({\frak g})$ of invariant complex structures on $M$, the Dolbeault cohomology of $M$ is isomorphic to the one of the differential bigraded algebra ass…

Spectral sequence connects knot homologies via algebraic geometry.

We generalize the Toda lattice hierarchy by considering N+M dependent variables. We construct roots and logarithms of the Lax operator which are uniquely defined operators with coefficients that are $ε$-series of differential polynomials in the dependent variables, and we use them to provide a Lax pair definition of th…

New algebra models refine complex manifold homotopy groups.

The paper surveys some new results and open problems connected with such fundamental combinatorial concepts as polytopes, simplicial complexes, cubical complexes, and subspace arrangements. Particular attention is paid to the case of simplicial and cubical subdivisions of manifolds and, especially, spheres. We describe…

Using a bigraded differential complex depending on the CR and pseudohermitian structure, we give a characterization of three-dimensional strongly pseudoconvex pseudo-hermitian CR-manifolds isometrically immersed in Euclidean space $\mathbb{R}^n$ in terms of an integral representation of Weierstrass type. Restricting to…

Khovanov homology is a bigraded Z-module that categorifies the Jones polynomial. The support of Khovanov homology lies on a finite number of slope two lines with respect to the bigrading. The Khovanov width is essentially the largest horizontal distance between two such lines. We show that it is possible to generate in…

New spectral invariant generalizes analytic torsion for manifolds with geometric product structure.

We study various aspects of the noncommutative residue for an algebra of pseudodifferential operators whose symbols have an expansion $a\sim \sum_{j=0}^\infty a_{m-j}, a_{m-j}(x,ξ)=\sum_{l=0}^k a_{m-j,l}(x,ξ) \log^l|ξ|,$ where $a_{m-j,l}$ is homogeneous in $ξ$ of degree $m-j$. We will explain why this algebra of pseudo…

For any graph G we define bigraded cohomology groups whose graded Euler characteristic is a multiple of the Yamada polynomial of G.

Studied knot Floer homology stabilization and prime mutant knots.

To each knot $K\subset S^3$ one can associated its knot Floer homology $\hat{HFK}(K)$, a finitely generated bigraded abelian group. In general, the nonzero ranks of these homology groups lie on a finite number of slope one lines with respect to the bigrading. The width of the homology is, in essence, the largest horizo…

Functoriality proved for colored link invariants.

For each graph, we construct a bigraded chain complex whose graded Euler characteristic is a version of the Tutte polynomial. This work is motivated by earlier work of Khovanov, Helme-Guizon and Rong, and others.

We prove that the foam and matrix factorization universal rational sl3 link homologies are naturally isomorphic as projective functors from the category of link and link cobordisms to the category of bigraded vector spaces.

The paper examines differential smoothness in specific algebra types.

This paper is concerned with nanowords, a generalization of links, introduced by Turaev. It is shown that the system of bigraded homology groups is an invariant of nanowords by introducing a new notion. This paper gives two examples which show the independence of this invariant from some of Turaev's homotopy invariants…

Determines algebra structure of complex differential forms operators.

The paper integrates DGLA to DGLG using HCPs and Hopf algebras.

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 determine the rational Khovanov bigraded homology groups of all Kanenobu knots. Also, we determine the crossing number for all Kanenobu knots $K(p,q)$ with $pq > 0$ or $|pq|\leq \max \{|p|, |q|\}$. In the case where $pq < 0$ and $|pq| > \max \{|p|, |q|\}$, we conjecture that the crossing number is $|p| + |q| + 8$.

To a presentation of an oriented link as the closure of a braid we assign a complex of bigraded vector spaces. The Euler characteristic of this complex (and of its triply-graded cohomology groups) is the HOMFLYPT polynomial of the link. We show that the dimension of each cohomology group is a link invariant.

The study examines differential smoothness in specific Artin-Schelter regular algebras of dimension 5.

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…

New algebraic formalism for differential calculus in Diolic algebras.

Develops differential K-theory for noncommutative algebras.

Study on smoothness of special algebra types.

This note explores two questions: (1) Which bigraded groups arise as the knot Floer homology of a knot in the three-sphere? (2) Given a knot, how many distinct knots share its Floer homology? Regarding the first, we show there exist bigraded groups satisfying all previously known constraints of knot Floer homology whic…

Lectures explore how differential methods improve understanding of algebraic group orbit spaces.

A new category of Lie algebras, called generalized Lie algebras, is presented such that classical Lie algebras and Lie-Rinehart algebras are objects of this new category. A new philosophy over generalized Lie algebroids theory is presented using the notion of generalized Lie algebra and examples of objects of the categ…

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 is to study 2-representations of 2-quantum gr…

Defines semi-symmetric metric connections on differential forms.

Researchers address the generation of differential invariants for geometric structures.

Smoothness of Sklyanin algebras examined in 3D and 4D cases.

Paper shows certain algebra types are not differentially smooth.

Study transcendence of abelian differential periods from bi-algebraic perspective.

Geometric AD framework simplifies derivative computation in JAX.

A general slice theorem for the action of a Fréchet Lie group on a Fréchet manifolds is established. The Nash-Moser theorem provides the fundamental tool to generalize the result of Palais to this infinite-dimensional setting. The presented slice theorem is illustrated by its application to gauge theories: the action o…