Researchers create spectral triples for twisted crossed products using Kasparov's external product.
problem Constructing spectral triples for twisted crossed products.
method Using Kasparov's external product, the construction of spectral triples for twisted crossed products is achieved.
result The construction of spectral triples for twisted crossed products is possible under suitable assumptions.
We develop a model characterizing all possible knots and links arising from recombination starting with a twist knot substrate, extending previous work of Buck and Flapan. We show that all knot or link products fall into three well-understood families of knots and links, and prove that given a positive integer n, the…
We study differential operators, whose coefficients define noncommutative algebras. As algebra of coefficients, we consider crossed products, corresponding to action of a discrete group on a smooth manifold. We give index formulas for Euler, signature and Dirac operators twisted by projections over the crossed product.…
Research decouples Lie algebroids using bicocycle double cross product theory.
problem Understanding decoupling and coupling phenomena in Lie algebroids.
method Bicocycle double cross product realization method.
result Unified product, double cross product, semi-direct product, and cocycle extension frameworks are instances of the general method.
This work merges 3-anchored bundles into 3-Lie algebroids.
problem Combining two 3-anchored bundles into a unified structure.
method Develops algebraic framework for merging bundles with mutual actions and cocycle terms.
result Unified setting for 3-Lie algebroids and special cases.
Calculates twist in Teichmüller space using cross ratios.
problem Calculating the Fenchel-Nielsen twist in Teichmüller space.
method Using cross ratio coordinates.
result Compact calculation of twist in Teichmüller space.
The paper studies knots formed by twisting a circle around a base knot and conjectures a linear growth in crossing numbers.
problem Understanding the growth rate of crossing numbers in twist families of knots.
method Introduced the stable crossing number and used geometric wrapping and algebraic winding to establish conjectures.
result The crossing number of Kn grows like nη(η−1) as no∞ for coherent twist families. The paper generalizes Kuperberg invariants using twisted Drinfeld doubles.
problem Quantum invariants of knots with additional structure.
method Using twisted Drinfeld doubles and Reshetikhin-Turaev invariants.
result Reidemeister torsion of knot complements as a quantum invariant.
Study shows fibred knots can't be untied with specific moves.
problem Untangling fibred knots with specific moves.
method Analyzes fibred knots and their untangling with $ar{t}_{2k}$-moves.
result Upper bound on two strand twist operations for untangling knots.
2-twist trefoil has 6 crossings, proving non-trivial knotted surface.
problem Computing the crossing number of non-trivial knotted surfaces.
method Bridge trisection and tri-plane diagrams to minimize crossings.
result 2-twist trefoil has crossing number 6.
The paper defines and analyzes curvature tensors on super twisted product spaces.
problem Investigating curvature tensors on super twisted product spaces.
method Defined W2-curvature tensor, computed curvature tensors and Ricci tensors, and studied curvature flatness. result Mixed Ricci-flat super twisted product semi-Riemannian manifolds can be expressed as super warped product manifolds.
The paper extends affine connection results to singular warped and twisted products.
problem Generalizing affine connections to singular warped and twisted products.
method Study of singular multiply warped products and singular twisted products with semi-symmetric metric and non-metric connections, discussing Koszul forms and curvature.
result Theoretical results on curvature and Koszul forms for singular multiply warped and twisted products.
Constructs an analytic index for infinite dimensional manifolds with LT-action.
problem Analyzing infinite dimensional manifolds with LT-action. method Defines an analytic LT-equivariant index using a Hilbert space, Dirac operator, and crossed product. result Justifies the analytic index in terms of noncommutative geometry.
Study examines properties of a specific type of geometric immersion.
problem Understanding geometric properties of doubly twisted product immersions.
method Established basic geometric properties through analysis.
result Basic geometric properties of doubly twisted product immersions have been identified.
In this article we define the twisted product of groups as the generalization of the semidirect product of groups. We will find the necessary and sufficient condition in order that the twisted product of groups to be a group. In particular, for two copies of the same group, the twisted product of group by itself throug…
Paper studies Landsberg curvature of a specific Finsler metric.
problem Analyzing Landsberg curvature of a particular Finsler metric.
method Derived Landsberg curvature and mean Landsberg curvature of the twisted product Finsler metric.
result Necessary and sufficient conditions for the metric to be Landsberg or weakly Landsberg are established.
In this paper, we compute the index form of the multiply twisted products. We study the Killing vector fields on the multiply twisted product manifolds and determine the Killing vector fields in some cases. We compute the curvature of the multiply twisted products with a semi-symmetric metric connection and show that t…
We prove that the property of admitting no cosmetic crossing changes is preserved under the operation of forming certain satellites of winding number zero. We also define strongly cosmetic crossing changes and we discuss their behavior under the operation of inserting full twists in the strings of closed braids.
Paper proves non-existence of certain Riemannian manifolds.
problem Non-existence of complete twisted and warped products.
method Proves Liouville-type theorems for Riemannian manifolds.
result Generalizes and complements results for compact manifolds.
On the product of two Finsler manifolds M1 M2, we consider the twisted metric F which is construct by using Finsler metrics F1 and F2 on the manifolds M1 and M2, respectively. We introduce horizontal and vertical distributions on twisted product Finsler manifold and study Creducible and semi-C-reducible properties of t…
Legendrian products can be twist spuns under certain conditions.
problem Understanding when Legendrian products are twist spuns.
method Using Legendrian contact homology and augmentation variety analysis.
result Not all Legendrian products are twist spuns.
Study on balanced Hermitian structures on Lie algebras twisted by representations.
problem Conditions for balanced and locally conformally balanced Hermitian structures on Lie algebras.
method Analysis of Hermitian structures on twisted cartesian products of Lie algebras.
result Classification of six-dimensional balanced Hermitian twisted cartesian products Lie algebras.
New racks defined; properties of rack representations explored.
problem Characterizing and studying rack representations.
method Definition of finitely stable racks, characterization of Alexander quandles, study of twisted rack dynamical systems, construction of cross-products, representation theory of racks and quandles.
result Definition and properties of Pontryagin dual of racks.
We show that if a knot admits a prime, twist-reduced diagram with at least 4 twist regions and at least 6 crossings per twist region, then every non-trivial Dehn filling of that knot is hyperbolike. A similar statement holds for links. We prove this using two arguments, one geometric and one combinatorial. The combinat…
We characterize Osserman and conformally Osserman Riemannian manifolds with the local structure of a warped product. By means of this approach we analyze the twisted product structure and obtain, as a consequence, that the only Osserman manifolds which can be written as a twisted product are those of constant curvature…
Study characterizes 2-Killing vector fields on complex spacetimes.
problem Characterize 2-Killing vector fields on multiply twisted product spacetimes. method Determine nonlinear differential equations, find twisted functions, provide solutions, and construct examples.
result Completely describe 2-Killing vector fields and twisted functions on multiply twisted product spacetimes. Improved bounds on volume-det inequality for links with many twists.
problem Vol-Det Conjecture for highly twisted alternating links.
method Improved Burton's and Stoimenow's bounds for links with more than eight twists.
result Enhanced inequalities for volume and determinant of links with many twists.
Given a 3-holed sphere decomposition of an orientable closed surface, it is shown that each orientation preserving homeomorphism of the surface is isotopic to a composition AB where A is a product of positive Dehn twists and B is a product of negative Dehn twists on the decomposition curves.
For 2+1 spacetime dimensions, we derive sufficient conditions for the twisting function in a twisted product spacetime, such that there is a global foliation by spacelike CMC surfaces.
In this paper, we show that the projection of a dualistic structure defined on a twisted product manifold induces dualistic structures on the base and the fiber manifolds, and conversely. Then under some conditions on the Ricci curvature and the Weyl conformal tensor we characterize dually flat structures on twisted pr…
New families of knots with more straight segments than crossings.
problem Understanding knots with more straight segments than crossings.
method Presented two families of knots, computed straight number for one, and provided a theorem about alternating knots.
result Explicit computation of straight number for one family of knots and a general theorem about alternating knots.
Develops index theory for infinite-dimensional manifolds with loop group actions using KK-theory.
problem Establish an index theory for infinite-dimensional manifolds acted upon by loop groups.
method Constructs objects in KK-theory and defines new cycles and indices.
result Defines a KK-theoretical index and compares it with the analytic index.
New method to untangle knots using null-homologous twists.
problem Finding the minimum number of twists to convert a knot to the unknot.
method Using null-homologous twists as a generalization of crossing changes.
result The untwisting number is at most twice the surgery description number plus 1.
This paper computes fixed point Floer cohomology for Dehn twists on surfaces.
problem Computing fixed point Floer cohomology for Dehn twists.
method Developed tools for computing fixed point Floer cohomology and product for Dehn twists in all dimensions.
result Splitting of the product and differential into local and Morse-theoretic contributions.
New method proves uniqueness of even plats diagrams.
problem Proving uniqueness of even plats diagrams.
method New arguments and restrictions on twist regions.
result Minimal width even plats diagrams are unique.
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.
The paper proves a theorem for a twisted Dirac operator on specific manifolds.
problem Analyzing the J-twist of the Dirac operator on spin manifolds.
method Lichnerowicz type formula and Kastler-Kalau-Walze type theorem for the J-twist of the Dirac operator.
result Proves a Kastler-Kalau-Walze type theorem for the J-twist of the Dirac operator on 3D and 4D almost product Riemannian spin manifolds with boundary.
This paper explores twisted Lagrangian tori in C^2 and their Hamiltonian stationarity.
problem Understanding the Hamiltonian stationarity of twisted Lagrangian tori in C^2.
method Investigation of differential geometry of twisted tori, including product and Chekanov's exotic tori.
result Only product tori are minimal under Hamiltonian deformations, indicating Chekanov's exotic tori are not area minimal.
We prove, under some mild conditions, that the equivariant twisted K-theory group of a crossed module admits a ring structure if the twisting 2-cocycle is 2-multiplicative. We also give an explicit construction of the transgression map T1:H∗(Γ;A)→H∗−1((N⋊Γ;A) for any crossed module N→Γ and prove…
Algorithm detects free products in disk mapping class groups.
problem Detecting free products in mapping class groups of punctured disks.
method Algorithm based on Dynnikov coordinates to verify completeness and reveal free product structure.
result Algorithm determines exact structure of free products generated by Dehn twists.
New rigidity results for warped product domains.
problem Scalar curvature rigidity of domains in warped products.
method Developed a new connection on a twisted spinor bundle and associated Dirac operator.
result Obtained Llarull type scalar curvature rigidity for a general class of domains in a warped product.
Twisted graph diagrams are virtual graph diagrams with bars on edges. A bijection between abstract graph diagrams and twisted graph diagrams is constructed. Then a polynomial invariant of Yamada-type is developed which provides a lower bound for the virtual crossing number of virtual graph diagrams.
The concordance orders of many algebraic order two knots of ten or fewer crossings have been heretofore unknown. We use Casson-Gordon invariants and twisted Alexander polynomials to find that, in all but one case, these knots do not have concordance order two. We also find that a certain family of algebraic order two t…
The Conway potential function (CPF) for colored links is a convenient version of the multi-variable Alexander-Conway polynomial. We give a skein characterization of CPF, much simpler than the one by Murakami. In particular, Conway's `smoothing of crossings' is not in the axioms. The proof uses a reduction scheme in a t…
We generalize the idea of unknotting knots to Seifert surfaces. We define an operation called ribbon twist which serves as the equivalent of a crossing change for knots. A Seifert surface is considered untwisted, the equivalent to unknotted, if it is isotopic to a standardly embedded n-fold punctured torus. A Seifert s…
The paper proves a theorem for a twisted Dirac operator on specific manifolds.
problem Analyzing the Dirac operator with torsion on spin manifolds.
method Develops a Lichnerowicz type formula and proves a Kastler-Kalau-Walze type theorem.
result Proves a Kastler-Kalau-Walze type theorem for the J-twist of the Dirac operator with torsion on 4D and 6D almost product Riemannian spin manifolds. Study shows how a strip's twisting increases at infinity, affecting its spectrum.
problem Understanding the spectrum of a strip with diverging twisting.
method Analyzing the Dirichlet Laplacian in a two-dimensional strip with segments rotating at increasing velocity.
result Essential spectrum forms a three-dimensional tube at infinity, with discrete eigenvalues possible if the tube's cross-section is a disk.
New relation found in 4D symplectic mapping class group.
problem Relation between Dehn twists in symplectic 4-manifolds.
method Holomorphic curve techniques, symplectic isotopy problem solution.
result Relation between two products of Dehn twists.