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.
New invariants measure entanglement of open curves in 3D space.
problem Characterizing the complexity of open curves in 3-space.
method Introducing linkoids and virtual knots, and new invariants.
result Strong invariants of linkoids independent of virtual closure.
Researchers extend Alexander polynomial to knotoids and linkoids.
problem Defining and studying Alexander polynomial extensions for knotoids and linkoids.
method Developed and proved conjecture on mock Alexander polynomial for knotoids and linkoids.
result Proved conjecture on mock Alexander polynomial for knotoids and linkoids.
Enhanced invariant for linkoids using quivers.
problem Counting invariants for linkoids.
method Use of quivers to generalize in-degree polynomial invariant.
result Introduced in-degree quiver polynomial matrix as a new invariant.
The paper defines and analyzes quandles of knotoids and linkoids.
problem Tackling the invariant properties of knotoids and linkoids.
method Defining fundamental quandles and pointed quandles, proving invariance, and introducing new invariants.
result Fundamental pointed quandles enhance the fundamental quandle and distinguish 1-linkoids.
Extends knotoid theory to multi-linkoids, studying various invariants.
problem Defining and analyzing invariants for multi-linkoids in a closed surface.
method Examines Kauffman bracket, ordered bracket, skein module, and T-invariant.
result Developed new invariants for multi-linkoids.
The paper defines new polynomials for links and linkoids.
problem No specific problem stated; focus on new polynomials.
method Defined as sums over states of link or linkoid diagrams with f=n. result Constructed new polynomials for starred links and linkoids.
Clock theorem extended to knotoids and linkoids.
problem Generalizing the Clock Theorem to knotoids and linkoids.
method Extending the Clock Theorem to knotoids and linkoids.
result Clock states of knotoid diagrams form a lattice under transpositions.
Proves hyperbolized groups are virtually compact special and linear.
problem Proving hyperbolized groups are virtually compact special and linear.
method Constructing an action of a hyperbolized group on a dual CAT(0) cubical complex.
result Proves hyperbolized groups are virtually compact special and linear.
The Thistlethwaite theorem is extended to knotoids and linkoids.
problem Extending a classical theorem to a new class of knots and links.
method Defining new invariants and associated polynomials for knotoids and linkoids.
result The Thistlethwaite theorem is proven for knotoids and linkoids.
New groups with special properties found.
problem Finding new groups with specific geometric properties.
method Proved actions on CAT(0) cubical complexes under certain conditions.
result Many groups admit cocompact actions on CAT(0) cubical complexes.
This paper gives a new interpretation of the virtual braid group in terms of a strict monoidal category SC that is freely generated by one object and three morphisms, two of the morphisms corresponding to basic pure virtual braids and one morphism corresponding to a transposition in the symmetric group. The key to this…
Novel Jones polynomial for open curves in 3D space.
problem Measuring entanglement complexity of open curves in 3-space.
method Defining Jones polynomial for linkoids and extending to collections of open and closed curves.
result Jones polynomial for open curves has real coefficients and is continuous.
Introduces XC-tangles for quantum tangle invariants.
problem Quantum invariants of tangles and virtual tangles.
method Introduces XC-tangles and their equivalence to virtual upwards tangles.
result Extension of knot isotopy invariants to XC-tangles.
A term structure model in which the short rate is zero is developed as a candidate for a theory of cryptocurrency interest rates. The price processes of crypto discount bonds are worked out, along with expressions for the instantaneous forward rates and the prices of interest-rate derivatives. The model admits function…
Paper studies knotoid chirality using shadow quandle colorings and invariants.
problem Distinguishing knotoids from their mirrors.
method Shadow quandle colorings and cocycle invariants.
result Knotoid 31 is shown to be chiral. The paper establishes conditions for strict power concavity in convolutions.
problem Conditions for strict power concavity in convolutions.
method Analyzes sufficient conditions for strict parabolic power concavity of convolutions.
result Establishes sufficient conditions for strict power concavity of convolutions.
Study strict stability of cones with isolated singularities.
problem Stability of cones with isolated singularities.
method Analyzes special Lagrangian and coassociative cones, provides examples for the complex case.
result Proves strict stability for special Lagrangian and coassociative cones.
We introduce a more restrictive version of the strict CD(K,∞) -condition, the so-called very strict CD(K,∞) -condition, and show the existence of optimal maps in very strict CD(K,∞) -spaces despite the possible lack of uniqueness of optimal plans.
Maps Lie 2-groups to Weil algebras, showing cohomology isomorphisms.
problem Cohomology of strict Lie 2-groups.
method Constructs van Est map using double complex and Weil algebra.
result Induces isomorphisms in cohomology under connectedness.
Proves integrability of strict Lie 2-algebras using cohomological methods.
problem Integrability of strict Lie 2-algebras.
method Van Est theorems relating cohomologies of Lie 2-groups and algebras.
result Proves integrability of Lie 2-algebras.
We study strict local martingales via h-transforms, a method which first appeared in Delbaen-Schachermayer. We show that strict local martingales arise whenever there is a consistent family of change of measures where the two measures are not equivalent to one another. Several old and new strict local martingales are i…
In an earlier work, we constructed the almost strict Morse n-category X which extends Cohen & Jones & Segal's flow category. In this article, we define two other almost strict n-categories V and W where V is based on homomorphisms between real vector spaces and $\ma…
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.
Study complex hyperbolic lattices and their relation to strict hyperbolization.
problem Understanding the relationship between complex hyperbolic lattices and strict hyperbolization.
method Analyzing the fundamental groups of complex hyperbolic manifolds and spaces arising from strict hyperbolization.
result Uniform lattices in PU(n,1) cannot be fundamental groups of Charney-Davis strict hyperbolizations when n ≥ 2.
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.
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.
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.
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.
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.
The virtual unknotting number of a virtual knot is the minimal number of crossing changes that makes the virtual knot to be the unknot, which is defined only for virtual knots virtually homotopic to the unknot. We focus on the virtual knot obtained from the standard (p,q)-torus knot diagram by replacing all crossings o…
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.
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.
In this paper, we initiate the study of a parametrised version of Rieffel's strict deformation quantization. We apply it to give a classification of noncommutative principal torus bundles, in terms of parametrised strict deformation quantization of ordinary principal torus bundles. The paper also contains a putative de…
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 …
We introduce a geometrically transparent strict saddle property for nonsmooth functions. This property guarantees that simple proximal algorithms on weakly convex problems converge only to local minimizers, when randomly initialized. We argue that the strict saddle property may be a realistic assumption in applications…
A virtual n-string is a chord diagram with n core circles and a collection of arrows between core circles. We consider virtual n-strings up to virtual homotopy, compositions of flat virtual Reidemeister moves on chord diagrams. Given a virtual 1-string α, Turaev associated a based matrix that encodes invariants…
New invariant for virtual n-links defined and studied.
problem Detecting virtual trefoil and other virtual links.
method Defining a new virtual link invariant VD(K) and studying its geometric properties. result The Dehn space DD(K) is an invariant of K and can detect the virtual trefoil. Given a virtual knot K, we construct a group VGK called the virtual knot group, and we use the elementary ideals of VGK to define invariants of K called the virtual Alexander invariants. For instance, associated to the k=0 ideal is a polynomial HK(s,t,q) in three variables which we call the virtual Alexa…
A virtual link diagram is called normal if the associated abstract link diagram is checkerboard colorable, and a virtual link is normal if it has a normal diagram as a representative. Normal virtual links have some properties similar to classical links.In this paper, we introduce a method of converting a virtual link d…
There are two major streams of literature on the modeling of financial bubbles: the strict local martingale framework and the Johansen-Ledoit-Sornette (JLS) financial bubble model. Based on a class of models that embeds the JLS model and can exhibit strict local martingale behavior, we clarify the connection between th…
New polynomial invariants for virtual links are stronger than F-polynomials.
problem Defining new invariants for virtual links.
method Introducing weight functions and a recurrent construction for new invariants.
result New polynomial invariants are stronger than F-polynomials.
New insights into matrix factorization show strict saddles have bounded eigenvalues.
problem Understanding the nature of critical points in matrix factorization.
method Analyzing orbits of critical points under the general linear group and identifying canonical points.
result Minimum eigenvalue of strict saddles is not uniformly bounded below zero.
Virtual singular braids are generalizations of singular braids and virtual braids. We define the virtual singular braid monoid via generators and relations, and prove Alexander- and Markov-type theorems for virtual singular links. We also show that the virtual singular braid monoid has another presentation with fewer g…
Paper detects checkerboard colorability of virtual links using odd writhe and arrow polynomial.
problem Detecting checkerboard colorability of virtual links.
method Using odd writhe and arrow polynomial.
result Proves 6 virtual knots are not checkerboard colorable.