arXiv research

Research decouples Lie algebroids using bicocycle double cross product theory.

This work merges 3-anchored bundles into 3-Lie algebroids.

We observe that the iterated tangent group of a Lie group may be realized as a double cross product of the 2nd order tangent group, with the Lie algebra of the base Lie group. Based on this observation, we derive the 2nd order Euler-Lagrange equations on the 2nd order tangent group from the 1st order Euler-Lagrange equ…

In this article, we will prove that the subsectors of $α$-induced sectors for $M \rtimes \hat{G} \supset M$ forms a modular category, where $M \rtimes \hat{G}$ is the crossed product of $M$ by the group dual $\hat{G}$ of a finite group $G$. In fact, we will prove that it is equivalent to Müger's crossed product. By usi…

Harmonic unit normal sections studied for Grassmannians induced by cross products.

We associate to each infinite primitive Lie pseudogroup a Hopf algebra of `transverse symmetries', by refining a procedure due to Connes and the first author in the case of the general pseudogroup. The affiliated Hopf algebra can be viewed as a `quantum group' counterpart of the infinite-dimensional primitive Lie algeb…

Traditionally, knot theorists have considered projections of knots where there are two strands meeting at every crossing. A triple crossing is a crossing where three strands meet at a single point, such that each strand bisects the crossing. In this paper we find a relationship between the triple crossing number and th…

Jones polynomial bounds and crossing numbers of knots.

The paper generalizes Kuperberg invariants using twisted Drinfeld doubles.

We show that for genus one knots the Alexander polynomial and the homology of the double cover branching over the knot provide obstructions to cosmetic crossings. As an application we prove the nugatory crossing conjecture for the negatively twisted, positive Whitehead doubles of all knots. We also verify the conjectur…

We study cosmetic crossings in knots of genus one and obtain obstructions to such crossings in terms of knot invariants determined by Seifert matrices. In particular, we prove that for genus one knots the Alexander polynomial and the homology of the double cover branching over the knot provide obstructions to cosmetic …

Serverless cloud computing speeds up double machine learning model estimation.

Study identifies prime strongly positive amphicheiral knots with double symmetry.

We translate into the double forms formalism the basic identities of Greub and Greub-Vanstone that were obtained in the mixed exterior algebra. In particular, we introduce a second product in the space of double forms, namely the composition product, which provides this space with a second associative algebra structure…

In this paper we present a certain class of geodesic vector fields of the double-twisted product R X R. Some examples of totally geodesic foliations are given.

Study Mazur doubles of knots and their relation to the Slope Conjecture.

In this paper, we discuss the crossing change operation along exchangeable double curves of a surface-knot diagram. We show that under certain condition, a finite sequence of Roseman moves preserves the property of those exchangeable double curves. As an application for this result, we also define a numerical invariant…

We construct a certain cross product of two copies of the braided dual $\tilde H$ of a quasitriangular Hopf algebra $H$, which we call the elliptic double $E_H$, and which we use to construct representations of the punctured elliptic braid group extending the well-known representations of the planar braid group attache…

A knot in the three-sphere is doubly slice if it is the cross-section of an unknotted two-sphere in the four-sphere. For low-crossing knots, the most complete work to date gives a classification of doubly slice knots through 9 crossings. We extend that work through 12 crossings, resolving all but four cases among the 2…

A conjecture proposed by J. Tripp in 2002 states that the crossing number of any knot coincides with the canonical genus of its Whitehead double. In the meantime, it has been established that this conjecture is true for a large class of alternating knots including $(2, n)$ torus knots, $2$-bridge knots, algebraic alter…

Researchers create spectral triples for twisted crossed products using Kasparov's external product.

Let K' be a knot that admits no cosmetic crossing changes and let C be a non-trivial, prime, non-cable knot. Then any knot that is a satellite of C with winding number zero and pattern K' admits no cosmetic crossing changes. As a consequence we prove the nugatory crossing conjecture for Whitehead doubles of prime, non-…

We show that for an alternating pretzel knot K the canonical genera of its Whitehead doubles W(K) are equal to the crossing number c(K) of K, verifying a conjecture of Tripp in the case of these knots.

Heegaard Floer homology connects to polynomial representations of Hecke algebras.

Derives equations for interacting Lie-Poisson systems using 2-cocycle extensions.

Defines cross product for m vectors in n-dimensional spaces.

Proves special alternating knots can't have cosmetic crossings.

A map from 3-manifold skein to Lagrangian skein via holomorphic curve counting.

Improved estimators for causal inference using cross-fitting and undersmoothing.

We will develop various methods, some are of geometric nature and some are of algebraic nature, to detect the various achiralities of knots and links in $S^3$. For example, we show that the twisted Whitehead double of a knot is achiral if and only if the double is the unknot or the figure eight knot, and we show that a…

Paper combines machine learning and model averaging for robust parameter estimation.

We construct branched double coverings by certain direct products of manifolds for connected sums of copies of sphere bundles over the 2-sphere. As an application we answer a question of Kotschick and Loeh up to dimension five. More precisely, we show that: (1) every simply connected, closed four-manifold admits a bran…

Given a standard complex semisimple Poisson Lie group $(G, π_{st})$, generalised double Bruhat cells $G^{u, v}$ and generalised Bruhat cells $O^u$ equipped with naturally defined holomorphic Poisson structures, where u, v are finite sequences of Weyl group elements, were defined and studied by Jiang Hua Lu and the auth…

The paper explores conditions for positive scalar curvature on manifolds with boundaries and their doubles.

We give an extension of Fox's formula of the Alexander polynomial for double branched covers over the three-sphere. Our formula provides the Reidemeister torsion of a double branched cover along a knot for a non-trivial one dimensional representation by the product of two factors derived from the knot group. One of the…

Previous work (Pradines, 1966, Aof and Brown, 1992) has given a setting for a holonomy Lie groupoid of a locally Lie groupoid. Here we develop analogous 2-dimensional notions starting from a locally Lie crossed module of groupoids. This involves replacing the Ehresmann notion of a local smooth coadmissible section of a…

Defines and extends flat pseudo-Riemannian F-Lie algebras.

Develops a test for conditional local independence of counting processes.

Alexander polynomial condition blocks crossing changes in some knots.

Double field theory was developed by theoretical physicists as a way to encompass $T$-duality. In this paper, we express the basic notions of the theory in differential-geometric invariant terms, in the framework of para-Kaehler manifolds. We define metric algebroids, which are vector bundles with a bracket of cross se…

In this article, we give a geometric proof of the classification of complex vector cross product due to Lee-Leung.

In this note a functorial approach to the integration problem of an LA-groupoid to a double Lie groupoid is discussed. To do that, we study the notions of fibred products in the categories of Lie groupoids and Lie algebroids, giving necessary and sufficient conditions for the existence of such. In particular, it turns …

The paper extends a theorem to Lie-Rinehart algebras and provides new decompositions of universal enveloping algebras.

We use the exterior product of double forms to reformulate celebrated classical results of linear algebra about matrices and bilinear forms namely the Cayley-Hamilton theorem, Laplace expansion of the determinant, Newton identities and Jacobi's formula for the determinant. This new formalism is then used to naturally g…

Let $G$ be a finite group. Noncommutative geometry of unital $G$-algebras is studied. A geometric structure is determined by a spectral triple on the crossed product algebra associated with the group action. This structure is to be viewed as a representative of a noncommutative orbifold. Based on a study of classical o…

Isomorphism found between filtered calculus and crossed products.

Uniform doubling property proven for specific Lie groups.

The study classifies immersed surfaces with knot group Z in simply-connected 4-manifolds.