arXiv research

Develops a new spectrum for annular links, recovering a transverse invariant at extreme gradings.

The paper studies knot Floer homology under Murasugi sum and establishes graded isomorphisms.

We prove that the spectrum constructed by González-Meneses, Manchón and the second author is stably homotopy equivalent to the Khovanov spectrum of Lipshitz and Sarkar at its extreme quantum grading.

Functor decomposes Khovanov spectra for non-alternating diagrams.

Generalizes pseudo-product structures with abnormal extremals.

The paper extends a geometric model using singular curves.

The Turaev genus of a link can be thought of as a way of measuring how non-alternating a link is. A link is Turaev genus zero if and only if it is alternating, and in this viewpoint, links with large Turaev genus are very non-alternating. In this paper, we study Turaev genus one links, a class of links which includes a…

Study detects a specific type of link using annular Khovanov homology.

Classifies states of four rebits using group theory.

This paper explores Khovanov adequacy in knot theory.

Contributions of recent deep-neural-network (DNN) based techniques have been playing a significant role in human-computer interaction (HCI) and user interface (UI) domains. One of the commonly used DNNs is human pose estimation. This kind of technique is widely used for motion capturing of humans, and to generate or mo…

Constructive approach to Lie algebra gradings, computing maximal and enumerating all gradings.

The geometry of graded principal bundles is discussed in the framework of graded manifold theory of Kostant-Berezin-Leites. In particular, we prove that a graded principal bundle is globally trivial if and only if it admits a global graded section and, further, that the sheaf of vertical derivations on such a bundle co…

Three definitions of graded vector bundles are shown to be equivalent.

Three new types of graded Lie groups are constructed and analyzed.

In this paper we discuss the question of integrating differential graded Lie algebras (DGLA) to differential graded Lie groups (DGLG). We first recall the classical problem of integration in the context, and present the construction for (non-graded) differential Lie algebras. Then, we define the category of differentia…

We review the concept of a graded bundle as a natural generalisation of a vector bundle. Such geometries are particularly nice examples of more general graded manifolds. With hindsight there are many examples of graded bundles that appear in the existing literature. We start with a discussion of graded spaces, passing …

This paper develops a theory of graded manifolds in differential geometry.

Graded Transformers embed algebraic structure in neural networks through graded transformations.

Combines generalized and graded geometry to explore new structures.

This paper aims at setting out the basics of $\mathbb{Z}$-graded manifolds theory. We introduce $\mathbb{Z}$-graded manifolds from local models and give some of their properties. The requirement to work with a completed graded symmetric algebra to define functions is made clear. Moreover, we define vector fields and ex…

The paper examines smoothness in graded skew Clifford algebras.

The paper defines Z-graded hom-Lie superalgebras and explores their properties.

In this paper, we construct a canonical grading on bordered Heegaard Floer homology by homotopy classes of nonvanishing vector fields. This grading is a generalization of our construction of an absolute grading on Heegaard Floer homology and it extends the well-known grading with values in a noncommutative group define…

A pseudo $H$-type Lie algebra naturally gives rise to a conformal pseudo-subriemannian fundamental graded Lie algebras. In this paper we investigate the prolongations of the associated fundamental graded Lie algebra and the associated conformal pseudo-subriemannian fundamental graded Lie algebra. In particular, we show…

Characterizes fundamental groups of disjointly tree-graded spaces.

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…

In this paper we first state the classification of the prolongations of complex free fundamental graded Lie algebras. Next we introduce the notion of free pseudo-product fundamental graded Lie algebras and study the prolongations of complex free pseudo-product fundamental graded Lie algebras. Furthermore we investigate…

Constructs graded jet bundles for Z-graded manifolds and vector bundles.

Extends manifold theory to $\mathcal{I}$-graded manifolds.

The paper investigates gradings of complex simple Lie algebras, focusing on $|3|$-gradings and their algebraic structures.

The paper studies graded manifolds and their functorial relationship.

Graded bundles are a particularly nice class of graded manifolds and represent a natural generalisation of vector bundles. By exploiting the formalism of supermanifolds to describe Lie algebroids we define the notion of a weighted $A$-connection on a graded bundle. In a natural sense weighted $A$-connections are adapte…

Study multiplicity-free covering of graded manifolds, proving equivalence of categories.

Heegaard Floer homology, first introduced by P. Ozsvath and Z. Szabo, associates to a 3-manifold Y a family of relatively graded Abelian groups HF(Y,t), indexed by Spin^c structures t on Y. In the case that Y is a rational homology sphere, Ozsvath and Szabo lift the relative Z-grading to an absolute Q-grading. This ind…

We study the notion of duality in the context of graded manifolds. For graded bundles, somehow like in the case of Gelfand representation and the duality: points vs. functions, we obtain natural dual objects which belongs to a different category than the initial ones, namely graded polynomial (co)algebra bundles and fr…

The abstract generalizes a construction for splitting supermanifolds and studies Lie supergroup cases.

New homology theory for graphs detects subdivisions and homology manifolds.

Study graded coverings for supermanifolds, proving their universal properties.

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…

We define \textit{graded manifolds} as a version of supermanifolds endowed with an additional $\mathbb Z$-grading in the structure sheaf, called \textit{weight} (not linked with parity). Examples are ordinary supermanifolds, vector bundles over supermanifolds, double vector bundles, iterated constructions like $TTM$, e…

In this paper we show how to recover the relative Q-grading in Heegaard Floer homology from the noncommutative grading on bordered Floer homology.

Peer grading is the process of students reviewing each others' work, such as homework submissions, and has lately become a popular mechanism used in massive open online courses (MOOCs). Intrigued by this idea, we used it in a course on algorithms and data structures at the University of Hamburg. Throughout the whole se…

Study bi-graded Lie algebras and their applications.

Abstract: Generalized reduction methods for symmetries in graded geometry.

Theory of $H$-graded manifolds and coverings of supermanifolds.

Paper adapts Getzler's grading technique for new applications.

In joint work with Yang Huang, we defined a canonical absolute grading on Heegaard Floer homology by homotopy classes of oriented 2-plane fields. A similar grading was defined on embedded contact homology by Michael Hutchings. In this paper we show that the isomorphism between these homology theories defined by Colin-G…