Study links' flat-virtual diagrams to create link invariants.
problem Equivalence and invariants of flat-virtual diagrams.
method Maps from links in thickened surfaces to flat-virtual links.
result Investigation of flat-virtual diagrams' equivalence and invariants.
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.
In this paper we investigate the virtual string links via a probabilistic interpretation. This representation can be used to distinguish some virtual string links from classical string links. In order to study the algebraic structure behind this probabilistic interpretation we introduce the notion of virtual flat biqua…
The paper introduces a semiquandle for flat virtual knots and connects it to u-polynomials.
problem Defining invariants for flat virtual knots.
method Developed a semiquandle and used it to define an invariant for flat virtual knots.
result The semiquandle invariant relates to u-polynomials for flat virtual knots.
We introduce an algebraic structure we call semiquandles whose axioms are derived from flat Reidemeister moves. Finite semiquandles have associated counting invariants and enhanced invariants defined for flat virtual knots and links. We also introduce singular semiquandles and virtual singular semiquandles which define…
The paper defines a chord index for virtual knots and links.
problem Defining a chord index for virtual knots and links.
method Smoothing real crossing points of virtual knot diagrams to construct a chord index.
result Several polynomial invariants can be derived from this construction.
In 2002, D. Hrencecin and L.H. Kauffman defined a filamentation invariant on oriented chord diagrams that may determine whether the corresponding flat virtual knot diagrams are non-trivial. A virtual knot diagram is non-classical if its related flat virtual knot diagram is non-trivial. Hence filamentations can be used …
This paper gives a polynomial invariant for flat virtual links. In the case of one component, the polynomial specializes to Turaev's virtual string polynomial. We show that Turaev's polynomial has the property that it is non-zero precisely when there is no filamentation of the knot, as described by Hrencecin and Kauffm…
New writhe polynomial for virtual links defined using weak chord index.
problem Defining a new writhe polynomial for virtual links.
method Constructing a weak chord index for self-crossing points and using it to define a new writhe polynomial.
result Three invariants of virtual links constructed using the new writhe polynomial.
We consider the group of unrestricted virtual braids, describe its structure and explore its relations with fused links. Also, we define the groups of flat virtual braids and virtual Gauss braids and study some of their properties, in particular their linearity.
The paper finds the minimum number of intersections for curves under virtual homotopy.
problem Finding the minimum number of intersections for curves under virtual homotopy.
method Using generalizations of the Anderson--Mattes--Reshetikhin Poisson bracket and the Cahn cobracket.
result The minimal number of intersections equals to the number of terms of a generalized Poisson bracket for pairs of curves, and can be counted by a generalized cobracket for a single curve.
In the present paper we give a new method for converting virtual knots and links to virtual braids. Indeed the braiding method given in this paper is quite general, and applies to all the categories in which braiding can be accomplished. We give a unifying topological interpretation of virtuals and flats (virtual strin…
In this paper, we define some polynomial invariants for virtual knots and links. In the first part we use Manturov's parity axioms to obtain a new polynomial invariant of virtual knots. This invariant can be regarded as a generalization of the odd writhe polynomial defined by the first author. The relation between this…
The paper constructs representations of flat virtual braids by free group automorphisms.
problem Representing flat virtual braids by automorphisms of free groups.
method Construction of representations of flat virtual braid groups FVBn by automorphisms of free groups of rank 2n. result Established conditions of faithfulness and properties of the kernel for n≥3. This paper defines a new invariant of virtual knots and links that we call the extended bracket polynomial, and denote by <<K>> for a virtual knot or link K. This invariant is a state summation over bracket states of the oriented diagram for K. Each state is reduced to a virtual 4-regular graph in the plane and the pol…
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…
Connected sum affects crossing numbers of flat virtual knots.
problem Understanding how connected sum impacts the crossing numbers of flat virtual knots.
method Analyzing minimal crossing diagrams and using super-additivity properties.
result Crossing number of flat virtual knots is super-additive under connected sum.
We describe a method of encoding various types of link diagrams, including those with classical, flat, rigid, welded, and virtual crossings. We show that this method may be used to encode link diagrams, up to equivalence, in a notation whose length is a cubic function of the number of 'riser marks'. For classical knots…
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.
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.
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.
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.
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.
Paper introduces flat-virtual knots and invariants for classical knots.
problem Constructing a map from classical knots to virtual knots.
method Definition of flat-virtual knots and invariants (Alexander-like polynomial, Kauffman bracket).
result Introduction of flat-virtual knots and their invariants.
Paper proves any twisted link can be described as a unique twisted braid.
problem Understanding twisted links and braids.
method Proved using Alexander and Markov Theorems for classical braids and links.
result Any twisted link can be described as the closure of a unique twisted braid.
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.
Extends classical results to virtual links, proving new properties of alternating and semi-alternating virtual links.
problem Classical results for virtual links, focusing on alternating and semi-alternating links.
method Inequality relating link determinant and crossing number, matrix-tree theorem, Tait conjectures for virtual and welded links.
result Alexander polynomial of almost classical alternating virtual links is alternating.
Wirtinger number equals virtual bridge number for virtual links.
problem Calculating the virtual bridge number of virtual links.
method Algorithmically computing the minimum number of generators of the link group.
result The Wirtinger number equals the virtual bridge number for virtual links.
New algebraic structure for distinguishing twisted virtual handlebody-links.
problem Distinguishing twisted virtual handlebody-links.
method Introducing twisted virtual bikeigebras and using them to define invariants.
result New invariants distinguish some twisted virtual handlebody-links.
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.In this paper, we introduce a method of converting a virtual link diagram to a normal virtual link diagram by use of the double covering …
Paper discusses when virtual links are equivalent as twisted links.
problem Determining equivalence of virtual links as twisted links.
method Using stable equivalence classes of links in oriented thickenings of surfaces.
result Necessary and sufficient condition for virtual links to be equivalent as twisted links.
Defines a new Steenrod square for virtual links, linking to Khovanov-Lipshitz-Sarkar stable homotopy type.
problem Studying Steenrod squares for virtual links.
method Defines a second Steenrod square for virtual links.
result First meaningful nontrivial example of the second Steenrod square on Khovanov homology.
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.
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. Paper constructs cyclic coverings of virtual link diagrams, proving equivalence of certain maps.
problem Constructing and proving equivalence of cyclic coverings of virtual link diagrams.
method Introducing m-fold cyclic covering diagrams and proving their equivalence to original diagrams. result Well-defined map from virtual links to mod m almost classical virtual links. We define a generalization of virtual links to arbitrary dimensions by extending the geometric definition due to Carter et al. We show that many homotopy type invariants for classical links extend to invariants of virtual links. We also define generalizations of virtual link diagrams and Gauss codes to represent virtua…
A virtual link can be understood as a link in a trivial I-bundle over an orientable compact surface with genus. A twisted virtual link is a link in a trivial I-bundle over a not-necessarily orientable compact surface. A twisted virtual birack is an algebraic structure with axioms derived from the twisted virtual Reidem…
Virtual knot theory, introduced by Kauffman, is a generalization of classical knot theory of interest because its finite-type invariant theory is potentially a topological interpretation of Etingof and Kazhdan's theory of quantization of Lie bi-algebras. Classical knots inject into virtual knots, and flat virtual knots…
The paper classifies virtual links using the arc shift operation.
problem Classifying \( n \)-component virtual links up to arc shift equivalence.
method Established the arc shift operation as an unknotting tool for \( n \)-homogeneous proper virtual links, explored its connection to the odd writhe, and identified sequences with specific arc shift bounds.
result Identified sequences of virtual link diagrams \( L_n \) with an upper bound of arc shift number equal to \( n \).
A new index measures how many changes are needed to turn virtual links into simple ones.
problem Determining the minimum number of changes needed to simplify virtual link diagrams.
method Defined the unknotting index using virtual crossings and classical changes, provided lower and upper bounds.
result Found bounds for the unknotting index of virtual links, including those from pretzel links.
Virtual links are hyperbolic with unique volume structure.
problem Extending hyperbolicity theory to virtual links.
method Using virtual link theory in thickened surfaces and totally geodesic boundaries.
result All virtual alternating links are tg-hyperbolic.
This paper classifies virtual string links up to cobordism using group theory.
problem Classifying virtual string links up to cobordism.
method Using group theory, specifically the group Zn(n−1). result Virtual string links up to cobordism are classified by elements of Zn(n−1). 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.
The paper confirms a conjecture and extends arrow polynomial to twisted links.
problem Extending classical invariants to virtual and twisted links.
method Checkerboard framings, cut points, and normalized arrow polynomial.
result The normalized arrow polynomial is an invariant for twisted links.
Classifies virtual links up to a specific move.
problem Characterizing 2-component virtual links using Ξ-moves. method Refined odd writhe and linking numbers.
result Classifies 2-component virtual links up to Ξ-moves. 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.
Minimal crossing virtual links have minimal supporting genus.
problem Understanding the relationship between the number of crossings and the genus of virtual links.
method Developed a new parity theory for virtual links to prove minimal crossing implies minimal genus.
result Minimal crossing virtual links have minimal supporting genus.
The notion of chckerboard colorability for virtual links and abstract links is introduced. We study the Jones polynomials of virtual links and abstruct links. It is proved that a certain property of the Jones polynomials of classical links is valid for virtual links which admit checkerboard colorings.