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.
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48 results for “virtualization moves”
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.
CF-moves for virtual linksmath.GT
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.
Classifies virtual links up to a specific move.
problem Characterizing -component virtual links using -moves.
method Refined odd writhe and linking numbers.
result Classifies -component virtual links up to -moves.
The H(n)-move simplifies virtual and welded knots and links.
problem Tackling the unknotting of virtual and welded links.
method Extending the H(n)-move to virtual and welded links and showing their equivalence to Reidemeister moves.
result Virtualization and forbidden move can be realized by a finite sequence of generalized Reidemeister moves and H(n)-moves.
The writhe polynomial invariant is proven for virtual knots via shell moves.
problem Determining equivalence of oriented virtual knots using writhe polynomials.
method Introducing shell moves to prove equivalence of writhe polynomials.
result Two virtual knots are equivalent if they can be transformed by shell moves.
New moves transform any virtual knot to a trivial knot.
problem Transforming virtual knots to trivial knots.
method Introducing arc shift and region arc shift moves.
result Any virtual knot can be transformed into a trivial knot using these 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.
This study simplifies verification of invariants in oriented virtual knots.
problem Verifying invariants of oriented virtual knots is complex and time-consuming.
method Identifying a minimal generating set of oriented virtual Reidemeister moves.
result A four-element subset serves as a generating set for oriented virtual Reidemeister moves.
Gordian complexes of knots and virtual knots given by region crossing changes and arc shift movesmath.GT
The paper explores Gordian complexes of knots and virtual knots using region crossing changes and arc shift moves.
problem Defining and analyzing Gordian complexes of knots and virtual knots using specific local moves.
method Region crossing change and arc shift move to construct Gordian complexes.
result Existence of arbitrarily high dimensional simplices in both Gordian complexes.
Virtual knots can be transformed by -moves, affecting their writhes.
problem Understanding transformations between virtual knots using -moves.
method Analyzing the effect of -moves on writhes and congruence classes.
result Two virtual knots related by -moves have congruent odd writhes modulo .
The paper discusses colorings and doubled colorings of virtual doodles.
problem Coloring virtual doodles using a new algebraic structure.
method Introduced a new algebra called a doodle switch and defined an invariant for virtual doodles.
result Introduced doubled colorings and defined an invariant for virtual doodles.
Up-down colorings of virtual-link diagrams and the necessity of Reidemeister moves of type IImath.GT
We introduce an up-down coloring of a virtual-link diagram. The colorabilities give a lower bound of the minimum number of Reidemeister moves of type II which are needed between two 2-component virtual-link diagrams. By using the notion of a quandle cocycle invariant, we determine the necessity of Reidemeister moves of…
The Jones polynomial's divisibility is analyzed via local moves on virtual links.
problem Divisibility of the Jones polynomial under local moves on virtual links.
method Developed a general decomposition for the Jones polynomial of virtual links and analyzed divisibility conditions for various local moves.
result Succinct divisibility conditions on the Jones polynomial of virtual links differing via local moves.
Forbidden detour moves unknot virtual knots.
problem Unknotting virtual knots using forbidden detour moves.
method Introduced forbidden detour moves as an unknotting operation for virtual knots and defined the forbidden detour number.
result Upper and lower bounds on the forbidden detour number are given in terms of knot invariants.
The paper studies generalized virtualization moves on welded links and classifies their equivalence.
problem Classifying equivalence of welded links under specific moves.
method Introduced and analyzed two new moves, and , on welded links.
result The -move is an unknotting operation for any , while is not for .
The forbidden moves can be combined with Gauss diagram Reidemeister moves to obtain move sequences with which we may change any Gauss diagram (and hence any virtual knot) into any other, including in particular the unknotted diagram
Two virtual link diagrams are homotopic if one may be transformed into the other by a sequence of virtual Reidemeister moves, classical Reidemeister moves, and self crossing changes. We recall the pure virtual braid group. We then describe the set of pure virtual braids that are homotopic to the identity braid.
New moves help untangle complex knots.
problem Deforming twisted knots into simpler forms.
method Finite sequences of extended Reidemeister moves and three forbidden moves.
result Any twisted knot can be simplified.
Forbidden moves categorify fused links into quivers.
problem Categorify fused links to invariant of links.
method Use forbidden moves to categorify fused links.
result Obtain three polynomial invariants of links.
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.
Proves Alexander- and Markov-type theorems for virtual trivalent braids.
problem Classifying virtual trivalent braids and graphs.
method Two versions of the Markov-type theorem: algebraic and L-move based.
result Established new theorems for virtual trivalent braids.
We define enhanced presentations of quandles via generators and relations with additional information comprising signed operations and an order structure on the set of generators. Such a presentation determines a virtual link diagram up to virtual moves. We list formal Reidemeister moves in which Tietze moves on the pr…
Classifies welded string links up to specific moves.
problem Classifying welded string links up to moves.
method Using Milnor invariants, classify welded string links up to -move and -move.
result Classifies welded string links up to -move and -move.
In this paper we prove a Markov Theorem for virtual braids and for some analogs of this structure. The virtual braid group is the natural companion in the category of virtual knots, just as the Artin braid group is the natural companion to classical knots and links. In this paper we follow the L--move methods to prove …
Parity on based matricesmath.GT
Parity defined for based matrices, a new example of virtual knot parity.
problem Defining parity for a new algebraic structure.
method Introduced parity for based matrices, defined reduced stable parity.
result New example of parity for virtual knots.
Invariant detects sliceness of virtual knots with specific chord indices.
problem Detecting sliceness in virtual knots with chord indices.
method Constructs an invariant using sliceness obstruction and sensitivity to Δ-move.
result Invariant detects sliceness and is sensitive to chord indices.
Graphoidsmath.CO
Graphoids are topological invariants of virtual graph diagrams.
problem Understanding knotted graphs with open ends in proteins and simplifying virtual spatial graphs.
method Topological interpretations of graphoids using graph Reidemeister moves.
result Virtual graphoids are useful for studying knotted graphs and simplifying spatial graphs.
On Free Knots and Linksmath.GT
Both classical and virtual knots arise as formal Gauss diagrams modulo some abstract moves corresponding to Reidemeister moves. If we forget about both over/under crossings structure and writhe numbers of knots modulo the same Reidemeister moves, we get a dramatic simplification of virtual knots, which kills all classi…
Study algebraic invariants of multi-virtual links.
problem Algebraic invariants of multi-virtual links.
method Determining moves, introducing operator quandles, constructing invariants.
result Established new invariants for multi-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.
This paper is a short introduction to and statement of the main theorems of our paper "Virtual Braids and the L-Move", JKTR, Vol. 15, No. 6 (2006), pp. 773-811. See also arxiv:Math.GT/0507035.
Every classical or virtual knot is equivalent to the unknot via a sequence of extended Reidemeister moves and the so-called forbidden moves. The minimum number of forbidden moves necessary to unknot a given knot is an invariant we call the {\it forbidden number}. We relate the forbidden number to several known invarian…
Virtual braids are a combinatorial generalization of braids. We present abstract braids as equivalence classes of braid diagrams on a surface, joining two distinguished boundary components. They are identified up to isotopy, compatibility, stability and Reidemeister moves. We show that virtual braids are in a bijective…
In the present paper, we consider local moves on classical and welded diagrams: (self-)crossing change, (self-)virtualization, virtual conjugation, Delta, fused, band-pass and welded band-pass moves. Interrelationship between these moves is discussed and, for each of these move, we provide an algebraic classification. …
Virtual Homotopymath.GT
Two welded (respectively virtual) link diagrams are homotopic if one may be transformed into the other by a sequence of extended Reidemeister moves, classical Reidemeister moves, and self crossing changes. In this paper, we extend Milnor's mu and bar mu invariants to welded and virtual links. We conclude this paper wit…
New invariants for welded and extended welded knots found using multi-skein relations.
problem Classifying and distinguishing welded and extended welded knots and links.
method Developed a multi-skein relation approach to find invariants for these knots and links.
result Found sufficient conditions on coefficients for invariance under extended Reidemeister moves.
Updated polynomial for virtual tangles, compatible with decompositions.
problem Generalizing polynomial for virtual tangles.
method Updated multi-variable affine index polynomial, introduced Turaev moves.
result Polynomial compatible with tangle decompositions and crossing weight recovery.
Rectangular mosaics extend virtual knot studies to larger polygons.
problem Studying virtual knots using mosaic techniques.
method Introduced rectangular mosaics, modified mosaic moves, and provided invariants.
result Developed algorithms for computing virtual knot invariants.
Innovative warping labeling for twisted knots and braids.
problem Inventing an invariant for twisted knots and braids.
method Introducing warping degree, constructing warping labeling, extending to virtual braids.
result Developed invariants for twisted knots and braids using warping labeling.
New invariants show stronger virtual knot sets.
problem Classifying virtual knots using invariants.
method Defining -order invariants using forbidden moves.
result Set of -order invariants is strictly stronger.
In this paper, it is shown that there are no nonconstant Goussarov-Polyak-Viro finite-type invariants that are invariant under the virtualization move. As an immediate corollary, we obtain the theorem which states none of the Birman coefficients of the Jones-Kauffman polynomial are of GPV finite type.
Khovanov and Rozansky's categorification of the HOMFLY-PT polynomial is invariant under braidlike isotopies for any link diagram and Markov moves for braid closures. To define HOMFLY-PT homology, they required a link to be presented as a braid closure, because they did not prove invariance under the other oriented Reid…
New invariant measures how many twists are needed to unknot welded knots.
problem Measuring complexity of welded knots.
method Local twist move, Alexander quandle coloring, Gordian distance.
result Established lower bound on twist number and related it to other knot invariants.
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 …
Multistring based matricesmath.GT
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.
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…
Virtual Knot Cobordismmath.GT
This paper defines a theory of cobordism for virtual knots and studies this theory for standard and rotational virtual knots and links. Non-trivial examples of virtual slice knots are given. Determinations of the four-ball genus of positive virtual knots are given using the results of a companion paper by the author an…