arXiv research

This paper reinterprets Khovanov-Sano symmetries using BV formalism.

Unified framework for Arnold-type invariants via dual complexes and finite-difference structures.

Study symmetries in equivariant Khovanov homology.

We show that over the binary field $\mathbb F_2$, the Bar-Natan perturbation of Khovanov homology splits as the direct sum of its two reduced theories, which we also prove are isomorphic. This extends Shumakovitch's analogous result for ordinary Khovanov homology, without the perturbation.

We review Bennequin type inequalities established using various versions of the Khovanov-Rozansky cohomology. Then we give a new proof of a Bennequin type inequality established by the author, and derive new Bennequin type inequalities for knots using Gornik's version of the Khovanov-Rozansky cohomology, which generali…

New proof of Khovanov-Rozansky homology base point independence in finite characteristic.

Lower bounds for a knot invariant are derived using computations and cobordism inequality.

Knots without 2-torsion have minimal Khovanov homology rank.

New integral expression quantizes Arnold strangeness.

We partially solve the conjecture by A.Shumakovitch about torsion in the Khovanov homology of prime, non-split links in S^3. We give a size restriction on the Khovanov homology of almost alternating links. We relate the Khovanov homology of the connected sum of a link diagram and the Hopf link with the Khovanov homolog…

The goal of this paper is to address A. Shumakovitch's conjecture about the existence of $\Z_2$-torsion in Khovanov link homology. We analyze torsion in Khovanov homology of semi-adequate links via chromatic cohomology for graphs which provides a link between the link homology and well-developed theory of Hochschild ho…

We present computational results about quasi-alternating knots and links and odd homology obtained by looking at link families in the Conway notation. More precisely, we list quasi-alternating links up to 12 crossings and the first examples of quasi-alternating knots and links with at least two different minimal diagra…

We are giving tables of quasi-alternating knots with $8\le n \le 12$ crossings. As the obstructions for a knot to be quasialternating we used homology thickness with regards to Khovanov homology, odd homology, and Heegaard-Floer homology $\widehat{HFK}$. Except knots which are homology thick, so cannot be quasialternat…

We show that the generalized Khovanov homology, defined by the second author in the framework of chronological cobordisms, admits a grading by the group $\mathbb{Z}\times\mathbb{Z}_2$, in which all homogeneous summands are isomorphic to the unified Khovanov homology defined over the ring $\mathbb{Z}_π:=\mathbb{Z}[π]/(π…

This paper explains how to compute Khovanov homology of torus links using the Kauffman bracket polynomial.

This paper completes the construction of arbitrary order conformally invariant differential operators in higher spin spaces. Jan Slovák has classified all conformally invariant differential operators on locally conformally flat manifolds. We complete his results in higher spin theory by giving explicit expressions for …

The paper connects eigenvalue problems for various operators and establishes inequalities and asymptotic formulas for heat traces.

Introduces a new elliptic operator with positive eigenvalue.

Proves Kato inequalities for various conformal operators.

We describe a set of conformally covariant boundary operators associated to the Paneitz operator, in the sense that they give rise to a conformally covariant energy functional for the Paneitz operator on a compact Riemannian manifold with boundary. These operators naturally give rise to a first- and third-order conform…

Study on biharmonic hypersurfaces with specific recurrent operators in Euclidean space.

Local index theorem for chiral geometric operators proved using heat kernel.

Study estimates eigenvalues for concave Hessian operators on convex domains.

The paper proves homotopy equivalences for spaces of unbounded Fredholm operators.

GJMS operators connect geometry, analysis, and physics.

Extends Calabi operator to Riemannian locally symmetric spaces.

New spectral torsion defined for rescaled Dirac operators.

Proves formal self-adjointness of certain differential operators.

The paper proves new theorems about specific types of operator perturbations.

Study of Dirac-like operators on spin manifolds with large mass parameters.

Study essential spectrum of differential operators on geometrically finite orbifolds.

The study proves inequalities for complex operators on curved spaces.

Study on opers over complex manifolds of dimension one.

Mixtures of neural operators reduce active complexity in operator learning.

Formula for Hadamard coefficients from Green's operators on spacetimes.

Study perturbs Dirac operators in any dimension, focusing on Majorana fermions.

Identifies Lorentzian locally symmetric spaces where Calabi operator suffices to determine Killing operator range.

Researchers create a family of conformally covariant operators.

The paper revisits and analyzes the tmd-operator in almost Kähler manifolds.

The k-Dirac operator is a differential operator which is natural to geometric structure of a parabolic type. We will give a set of initial conditions for this operator. In the proof of the claim we will need to adapt some parts from the theory of exterior differential systems to the setting of weighted differential ope…

We describe a set of conformally covariant boundary operators associated to the sixth-order GJMS operator on a conformally invariant class of manifolds which includes compactifications of Poincaré--Einstein manifolds. This yields a conformally covariant energy functional for the sixth-order GJMS operator on such manifo…

The paper derives expansions for Green's operators and resolvents using Hadamard methods.

The paper quantizes Kähler manifolds using differential operators.

The paper classifies and proves properties of symmetry breaking operators for specific groups.

New operators generalize Michelsohn's on almost Hermitian manifolds.

Paper quantifies neural operators' efficiency for solving nonlinear parabolic PDEs.

Researchers describe local properties of Haantjes operators.

We use the spectra of Dirac type operators on the sphere $S^{n}$ to produce sharp $L^{2}$ inequalities on the sphere. These operators include the Dirac operator on $S^{n}$, the conformal Laplacian and Paenitz operator. We use the Cayley transform, or stereographic projection, to obtain similar inequalities for powers o…