Defines virtual immersions to characterize symmetric spaces.
problem Characterizing symmetric spaces using virtual immersions.
method Defines virtual immersions as a generalization of isometric immersions, proves characterization of symmetric spaces.
result A manifold admits a virtual immersion with skew symmetric second fundamental form if and only if it is a symmetric space.
The paper introduces the spirality character of the almost fiber part for a closed essentially immersed subsurface of a closed orientable aspherical 3-manifold, which generalizes an invariant due to Rubinstein and Wang. The subsurface is virtually embedded if and only if the almost fiber part is aspiral, and in this ca…
Proves one-relator groups with negative immersions are hyperbolic and virtually special.
problem One-relator groups with negative immersions.
method Refinement of Magnus--Moldavanskii hierarchy and introduction of Z-stable HNN-extensions and hierarchies.
result One-relator groups with negative immersions are hyperbolic and virtually special, resolving a conjecture.
Proves Alexander and Markov theorems for higher genus virtual doodles.
problem Classifying isotopy classes of immersed circles on surfaces.
method Introduces virtual twin groups to extend twin groups and prove theorems for higher genus.
result Proves Alexander and Markov theorems for higher genus virtual doodles.
A hyperbolic conjugacy class in the modular group PSL(2,Z) corresponds to a closed geodesic in the modular orbifold. Some of these geodesics virtually bound immersed surfaces, and some do not; the distinction is related to the polyhedral structure in the unit ball of the stable commutator length norm. We prove the foll…
Distorted surfaces in graph manifolds have specific distortion properties.
problem Distortion of surfaces in graph manifolds.
method Analysis of immersed horizontal surfaces in 3D graph manifolds.
result Fundamental group of surfaces is quadratically distorted if virtually embedded, exponentially distorted otherwise.
We introduce a new cohomology-theoretic method for classifying generic immersed curves in closed compact surfaces by using Gauss codes. This subsumes a result of J.S. Carter on classifying immersed curves in oriented compact surfaces, and provides a criterion for when an immersion is two-colorable. We note an applicati…
We show that many 3-manifold groups have no nonabelian surface subgroups. For example, any link of an isolated complex surface singularity has this property. In fact, we determine the exact class of closed graph-manifolds which have no immersed pi_1-injective surface of negative Euler characteristic. We also determine …
We address the question of detecting minimal virtual diagrams with respect to the number of virtual crossings. This problem is closely connected to the problem of detecting the minimal number of additional intersection points for a generic immersion of a singular link in R2. We tackle this problem by the so-called…
We consider the following properties of compact oriented irreducible graph-manifolds: to contain a π1-injective surface (immersed, virtually embedded or embedded), be (virtually) fibered over S1, and to carry a metric of nonpositive sectional curvature. It turns out that all these properties can be described from…
We show that every closed, virtually fibered hyperbolic 3-manifold contains immersed, quasi-Fuchsian surfaces with convex cores of arbitrarily large thickness.
Paper introduces skew-symmetric matrices for virtual doodle classification.
problem Classifying virtual doodles and distinguishing non-classical doodles.
method Use skew-symmetric augmented matrices and homology intersection number.
result Characterization of virtualization of classical doodles.
The paper develops a method to map knots in a cylinder to virtual-flat knots.
problem How to map knots in a cylinder to virtual knots.
method Construct a diagram on a cylinder with invisible crossings, then pull back invariants.
result Developed a method to map knots in a cylinder to virtual-flat knots.
We prove that in any hyperbolic orbifold with one boundary component, the product of any hyperbolic fundamental group element with a sufficiently large multiple of the boundary is represented by a geodesic loop that virtually bounds an immersed surface. In the case that the orbifold is a disk, there are some conditions…
Paper defines doodles on closed surfaces, unifying classical and virtual theories.
problem Classifying doodles on closed surfaces, especially non-orientable ones.
method Introducing twisted virtual doodles, defining twin groups, and proving Alexander- and Markov-type theorems.
result Unified theory of doodles, showing trivial center and residually finite properties.
We show that for certain hyperbolic 3-manifolds, all boundary slopes are slopes of immersed incompressible surfaces, covered by incompressible embeddings in some finite cover. The manifolds include hyperbolic punctured torus bundles and hyperbolic two-bridge knots.
Virtual twin groups map to symmetric groups, revealing automorphism structure.
problem Understanding homomorphisms between virtual twin groups and symmetric groups.
method Using irreducible right-angled Coxeter groups and right-angled Artin groups.
result A complete description of homomorphisms between virtual twin groups and symmetric groups, including the structure of the automorphism group of VTn. Study finds minimal hypersurfaces in compact symmetric spaces have a linear index bound.
problem Understanding minimal hypersurfaces in compact symmetric spaces.
method Introduced a generalized isometric immersion structure for compact symmetric spaces, proving bounds on the index and nullity.
result Minimal hypersurfaces in compact symmetric spaces have a linear index bound, depending on the first Betti number.
Study uses sentiment analysis to predict cryptocurrency token returns in virtual reality.
problem Predicting cryptocurrency token returns in virtual reality economies.
method Used BERT for sentiment analysis and developed LSTM models integrating multi-modal features.
result Multi-modal model significantly outperforms price-only baseline in prediction accuracy.
VR enables professionals to develop deep learning models by moving virtual objects.
problem Challenges in understanding and developing deep learning models.
method Built a VR-based DL development environment where users interact with tangible objects to construct neural networks.
result Users can develop and understand DL models intuitively through VR, with real-time accuracy feedback.
Paper studies pure virtual twin groups and their automorphisms.
problem Understanding the structure and automorphisms of pure virtual twin groups.
method Analyzes stable isotopy classes of immersed circles on surfaces, extending classical knot theory.
result Proves PVTn is an irreducible right-angled Artin group with trivial center and gives its precise presentation. Virtual reality brings non-Euclidean geometry to life.
problem Understanding non-Euclidean geometry is challenging.
method Interactive visualizations in virtual reality.
result Users can experience non-Euclidean geometry firsthand.
Doodles on surfaces expanded from 2-sphere to any closed orientable surface.
problem Extending doodles from 2-sphere to any closed orientable surface.
method Introduced and proved uniqueness of minimal representatives, gave examples, and introduced virtual doodles.
result Natural one-to-one correspondence between doodles on surfaces and virtual doodles on the plane.
We give a quantification of residual finiteness for the fundamental groups of hyperbolic manifolds that admit a totally geodesic immersion to a compact, right-angled Coxeter orbifold of dimension 3 or 4. Specifically, we give explicit upper bounds on residual finiteness that are linear in terms of geodesic length. We t…
Proves conditions for nearby special Lagrangians in Calabi-Yau manifolds.
problem Conditions for nearby special Lagrangians in Calabi-Yau manifolds.
method Analyzes fundamental group, Floer cohomology, and special Lagrangian conditions.
result Conditions for nearby special Lagrangians in Calabi-Yau manifolds.
Ribbon 2-knotted objects are locally flat embeddings of surfaces in 4-space which bound immersed 3-manifolds with only ribbon singularities. They appear as topological realizations of welded knotted objects, which is a natural quotient of virtual knot theory. In this paper we consider ribbon tubes and ribbon torus-link…
GANs improve building performance model accuracy by integrating occupant behaviors.
problem Discrepancies between design and operation performance in buildings.
method Generative Adversarial Networks (GANs) to learn mixture models combining existing BPMs with occupant behaviors.
result Augmented BPMs significantly outperform existing BPMs in achieving specified performance targets.
This paper improves route choice models by incorporating contextual factors.
problem Existing route choice models lack consideration of dynamic contextual conditions.
method Knowledge distillation from Stated Choice Experiments in Immersive Virtual Environment.
result High-fidelity route choice models with increased predictive power.
In the present paper, we construct a simple invariant which provides a sliceness obstruction for {\em free knots}. This obstruction provides a new point of view to the problem of studying cobordisms of curves immersed in 2-surfaces, a problem previously studied by Carter, Turaev, Orr, and others. The obstruction to sli…
Deep learning framework detects emotions from EEG data.
problem Detecting emotions from EEG signals.
method Temporal and spatial convolutional layers learn discriminative representations.
result TSception achieves 86.03% classification accuracy, significantly outperforming other methods.
New virtualized Δ-move simplifies virtual knots and links.
problem Simplifying virtual knots and links.
method Introducing a new local deformation called the virtualized Δ-move.
result Virtualized Δ-move is an unknotting operation for virtual knots.
Virtual index cocycles reformulate virtual link invariants.
problem No specific problem stated; focuses on reformulation.
method Using virtual index cocycles to reformulate invariants.
result Unified reformulation of virtual link invariants.
The paper studies strict equivalence in multi-virtual linkoids with new invariants.
problem Understanding strict equivalence in multi-virtual linkoids.
method Utilizing multi-virtual knot theory, defining strict virtual linkoids, and studying invariants.
result New invariants for strict virtual linkoids are defined.
Paper introduces a new invariant for planar knotoids.
problem Defining an invariant for planar knotoids.
method Using Gauss diagrams and transcendental functions.
result The invariant is a Vassiliev invariant of order one.
The study enumerates virtual quandles up to isomorphism.
problem Classifying virtual quandles up to isomorphism.
method Computer search and classification based on conjugacy class structures of rack automorphism groups.
result Classifications of virtual racks and quandles up to order 8.
A new method for virtual homotopy classes of virtual strings.
problem Addressing an open problem of Turaev regarding virtual homotopy classes of virtual strings.
method Generalizing Turaev's method to multistring based matrices for virtual n-strings.
result Construction of similar invariants for virtual homotopy classes of virtual n-strings.
The paper extends CF-moves to classify virtual links of any number of components.
problem Classifying virtual links using CF-moves.
method Extending CF-moves to classify virtual links of arbitrary number of components using the virtual linking number and invariants.
result Classification of 3-component even virtual links up to CF-moves.
New virtual version of Thompson's group created to handle virtual knots.
problem Creating a mathematical framework for virtual knots.
method Defining a virtual version of Thompson's group and proving its properties.
result Any virtual link can be constructed from an element of the new virtual Thompson's group.
Refines virtual link equality criterion for diagrams with one virtual crossing.
problem Determining when a virtual link diagram represents a properly virtual link.
method Refines the Kauffman-Murasugi-Thislethwaite type inequality for virtual links.
result Criterion for virtual link diagrams with exactly one virtual crossing to represent a properly virtual link.
Study shows unknotting number of certain virtual torus knots equals standard torus knot's unknotting number.
problem Determining the unknotting number of virtual torus knots.
method Analyzing virtual knots derived from standard torus knots and counting crossing changes.
result Virtual unknotting number of certain virtual torus knots equals the unknotting number of the corresponding standard torus knot.
Extend writhe polynomial from virtual knots to multi-virtual knots
problem Extend writhe polynomial from virtual knots to multi-virtual knots
method Extend writhe polynomial from virtual knots to multi-virtual knots
result Several questions asked in [16] have been answered
New invariants for virtual knots and links defined via quiver representations.
problem Defining new invariants for virtual knots and links.
method Quiver representations associated to virtual biquandles and rings.
result New polynomial invariants for virtual knots and links.
Study virtualized Delta, Sharp, and Pass moves for oriented virtual knots and links.
problem Conditions for unknotting oriented virtual knots and links.
method Local moves (Delta, Sharp, Pass) for oriented virtual links.
result Lower bounds for the unknotting numbers are proven and shown to be best possible.
This paper connects virtual biquandles to biquandles for virtual link colorings.
problem Extending invariants from biquandles to virtual biquandles.
method Establishing equivalence between two representations of virtual braid groups and introducing new labeling rules.
result The number of colorings of a virtual link by virtual biquandles can be recovered from colorings by biquandles.
Method converts virtual link diagrams to normal form, preserving equivalence.
problem Normalizing virtual link diagrams to study their properties.
method Method converts virtual link diagrams to normal form.
result Normal virtual link diagrams from equivalent diagrams are equivalent.
Virtual knots and links get multicrossings and petal diagrams.
problem Defining multicrossings for virtual knots and links.
method Extending multicrossings to virtual knots and links, showing petal diagrams exist.
result Petal diagrams exist for all virtual knots.
Introduces virtual tribrackets for knot coloring.
problem Coloring regions in virtual knot diagrams.
method Algebraic structure (virtual tribrackets) for knot coloring.
result Invariant can distinguish certain virtual knots.
A virtual string can be defined as an equivalence class of planar diagrams under certain kinds of diagrammatic moves. Virtual strings are related to virtual knots in that a simple operation on a virtual knot diagram produces a diagram for a virtual string. In this paper we consider three operations on a virtual string …