scholarly journals Rotational virtual knots and quantum link invariants

2015 ◽  
Vol 24 (13) ◽  
pp. 1541008 ◽  
Author(s):  
Louis H. Kauffman
Keyword(s):  
Knot Theory ◽  
Link Invariant ◽  
Quantum Invariants ◽  
Virtual Knot ◽  
Link Invariants ◽  
Virtual Knots ◽  
Virtual Knot Theory ◽  
Quantum Link ◽  
Knots And Links

This paper studies rotational virtual knot theory and its relationship with quantum link invariants. Every quantum link invariant for classical knots and links extends to an invariant of rotational virtual knots and links. We give examples of non-trivial rotational virtuals that are undectable by quantum invariants.

scholarly journals CLASSIFICATION OF FLAT VIRTUAL PURE TANGLES

2013 ◽  
Vol 22 (04) ◽  
pp. 1340006
Author(s):  
KARENE CHU
Keyword(s):  
Invariant Theory ◽  
Knot Theory ◽  
Virtual Link ◽  
Lie Bialgebras ◽  
Virtual Knot ◽  
Virtual Knots ◽  
High Algebra ◽  
Virtual Knot Theory ◽  
Knots And Links

Virtual knot theory, introduced by Kauffman [Virtual Knot theory, European J. Combin.20 (1999) 663–690, arXiv:math.GT/9811028], is a generalization of classical knot theory of interest because its finite-type invariant theory is potentially a topological interpretation [D. Bar-Natan, u, v, w-knots: Topology, Combinatorics and low and high algebra] of Etingof and Kazhdan's theory of quantization of Lie bi-algebras [Quantization of Lie Bialgebras, I, Selecta Math. (N.S.) 2 (1996) 1–41, arXiv:q-alg/9506005]. Classical knots inject into virtual knots [G. Kuperberg, What is Virtual Link? Algebr. Geom. Topol.3 (2003) 587–591, arXiv:math.GT/0208039], and flat virtual knots [V. O. Manturov, On free knots, preprint (2009), arXiv:0901.2214; On free knots and links, preprint (2009), arXiv:0902.0127] is the quotient of virtual knots which equates the real positive and negative crossings, and in this sense is complementary to classical knot theory within virtual knot theory. We completely classify flat virtual tangles with no closed components (pure tangles). This classification can be used as an invariant on virtual pure tangles and virtual braids.

ON A LOCAL MOVE FOR VIRTUAL KNOTS AND LINKS

2009 ◽  
Vol 18 (11) ◽  
pp. 1577-1596 ◽  
Author(s):  
TOSHIYUKI OIKAWA
Keyword(s):  
Sufficient Conditions ◽  
Virtual Link ◽  
Link Invariant ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Linking Number ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Necessary And Sufficient ◽  
Knots And Links

We define a local move called a CF-move on virtual link diagrams, and show that any virtual knot can be deformed into a trivial knot by using generalized Reidemeister moves and CF-moves. Moreover, we define a new virtual link invariant n(L) for a virtual 2-component link L whose virtual linking number is an integer. Then we give necessary and sufficient conditions for two virtual 2-component links to be deformed into each other by using generalized Reidemeister moves and CF-moves in terms of a virtual linking number and n(L).

scholarly journals FIBERED KNOTS AND VIRTUAL KNOTS

2013 ◽  
Vol 22 (12) ◽  
pp. 1341003 ◽  
Author(s):  
MICAH W. CHRISMAN ◽  
VASSILY O. MANTUROV
Keyword(s):  
Knot Theory ◽  
Basic Theory ◽  
Covering Space ◽  
New Technique ◽  
Regular Neighborhood ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Fibered Knots ◽  
A New Technique ◽  
Virtual Knot Theory

We introduce a new technique for studying classical knots with the methods of virtual knot theory. Let K be a knot and J be a knot in the complement of K with lk (J, K) = 0. Suppose there is covering space [Formula: see text], where V(J) is a regular neighborhood of J satisfying V(J) ∩ im (K) = ∅ and Σ is a connected compact orientable 2-manifold. Let K′ be a knot in Σ × (0, 1) such that πJ(K′) = K. Then K′ stabilizes to a virtual knot [Formula: see text], called a virtual cover of K relative to J. We investigate what can be said about a classical knot from its virtual covers in the case that J is a fibered knot. Several examples and applications to classical knots are presented. A basic theory of virtual covers is established.

scholarly journals A twisted link invariant derived from a virtual link invariant

2017 ◽  
Vol 26 (09) ◽  
pp. 1743007
Author(s):  
Naoko Kamada
Keyword(s):  
Knot Theory ◽  
Orientable Surface ◽  
Link Diagram ◽  
Link Invariant ◽  
Virtual Knot ◽  
Chord Diagrams ◽  
Link Diagrams ◽  
Virtual Links ◽  
Virtual Knot Theory ◽  
Double Coverings

Virtual knot theory is a generalization of knot theory which is based on Gauss chord diagrams and link diagrams on closed oriented surfaces. A twisted knot is a generalization of a virtual knot, which corresponds to a link diagram on a possibly non-orientable surface. In this paper, we discuss an invariant of twisted links which is obtained from the JKSS invariant of virtual links by use of double coverings. We also discuss some properties of double covering diagrams.

scholarly journals GRAPH-VALUED INVARIANTS OF VIRTUAL AND CLASSICAL LINKS AND MINIMALITY PROBLEM

2013 ◽  
Vol 22 (12) ◽  
pp. 1341006 ◽  
Author(s):  
VLADIMIR ALEKSANDROVICH KRASNOV ◽  
VASSILY OLEGOVICH MANTUROV
Keyword(s):  
Knot Theory ◽  
Classical Case ◽  
Strong Sense ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Crucial Difference ◽  
Virtual Links ◽  
Two Component ◽  
The Rich ◽  
Virtual Knot Theory

The Kuperberg bracket is a well-known invariant of classical links. Recently, the second named author and Kauffman constructed the graph-valued generalization of the Kuperberg bracket for the case of virtual links: unlike the classical case, the invariant in the virtual case is valued in graphs which carry a significant amount of information about the virtual knot. The crucial difference between virtual knot theory and classical knot theory is the rich topology of the ambient space for virtual knots. In a paper by Chrisman and the second named author, two-component classical links with one fibered component were considered; the complement to the fibered component allows one to get highly non-trivial ambient topology for the other component. In this paper, we combine the ideas of the above mentioned papers and construct the "virtual" Kuperberg bracket for two-component links L = J ⊔ K with one component (J) fibered. We consider a new geometrical complexity for such links and establish minimality of diagrams in a strong sense. Roughly speaking, every other "diagram" of the knot in question contains the initial diagram as a subdiagram. We prove a sufficient condition for minimality in a strong sense where minimality cannot be established as introduced in the paper by Chrisman and the second named author.

scholarly journals Alexander and writhe polynomials for virtual knots

2016 ◽  
Vol 25 (08) ◽  
pp. 1650050 ◽  
Author(s):  
Blake Mellor
Keyword(s):  
Knot Theory ◽  
Alexander Polynomial ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Virtual Links ◽  
New Interpretation ◽  
Polynomial Formula ◽  
Knots And Links

We give a new interpretation of the Alexander polynomial [Formula: see text] for virtual knots due to Sawollek [On Alexander–Conway polynomials for virtual knots and Links, preprint (2001), arXiv:math/9912173] and Silver and Williams [Polynomial invariants of virtual links, J. Knot Theory Ramifications 12 (2003) 987–1000], and use it to show that, for any virtual knot, [Formula: see text] determines the writhe polynomial of Cheng and Gao [A polynomial invariant of virtual links, J. Knot Theory Ramifications 22(12) (2013), Article ID: 1341002, 33pp.] (equivalently, Kauffman’s affine index polynomial [An affine index polynomial invariant of virtual knots, J. Knot Theory Ramifications 22(4) (2013), Article ID: 1340007, 30pp.]). We also use it to define a second-order writhe polynomial, and give some applications.

scholarly journals POLYNOMIAL KNOT AND LINK INVARIANTS FROM THE VIRTUAL BIQUANDLE

2013 ◽  
Vol 22 (04) ◽  
pp. 1340004 ◽  
Author(s):  
ALISSA S. CRANS ◽  
ALLISON HENRICH ◽  
SAM NELSON
Keyword(s):  
Polynomial Ring ◽  
Laurent Polynomial ◽  
Alexander Polynomial ◽  
Virtual Knot ◽  
Link Invariants ◽  
Virtual Knots ◽  
Monomial Ordering ◽  
Laurent Polynomial Ring ◽  
Future Work ◽  
Knots And Links

The Alexander biquandle of a virtual knot or link is a module over a 2-variable Laurent polynomial ring which is an invariant of virtual knots and links. The elementary ideals of this module are then invariants of virtual isotopy which determine both the generalized Alexander polynomial (also known as the Sawollek polynomial) for virtual knots and the classical Alexander polynomial for classical knots. For a fixed monomial ordering <, the Gröbner bases for these ideals are computable, comparable invariants which fully determine the elementary ideals and which generalize and unify the classical and generalized Alexander polynomials. We provide examples to illustrate the usefulness of these invariants and propose questions for future work.

LONG VIRTUAL KNOTS AND THEIR INVARIANTS

2004 ◽  
Vol 13 (08) ◽  
pp. 1029-1039 ◽  
Author(s):  
VASSILY O. MANTUROV
Keyword(s):  
Knot Theory ◽  
Classical Case ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Virtual Links ◽  
Virtual Knot Theory

There are some phenomena arising in the virtual knot theory which are not the case for classical knots. One of them deals with the "breaking" procedure of knots and obtaining long knots. Unlike the classical case, they might not be the same. The present work is devoted to construction of some invariants of long virtual links. Several explicit examples are given. For instance, we show how to prove the non-triviality of some knots obtained by breaking virtual unknot diagrams by very simple means.

ABSTRACT LINK DIAGRAMS AND VIRTUAL KNOTS

2000 ◽  
Vol 09 (01) ◽  
pp. 93-106 ◽  
Author(s):  
Naoko KAMADA ◽  
Seiichi KAMADA
Keyword(s):  
Knot Theory ◽  
Geometric Interpretation ◽  
Equivalence Classes ◽  
Virtual Link ◽  
Link Diagram ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Link Diagrams ◽  
Higher Dimensional ◽  
Virtual Knot Theory

The notion of an abstract link diagram is re-introduced with a relationship with Kauffman's virtual knot theory. It is prove that there is a bijection from the equivalence classes of virtual link diagrams to those of abstract link diagrams. Using abstract link diagrams, we have a geometric interpretation of the group and the quandle of a virtual knot. A generalization to higher dimensional cases is introduced, and the state-sum invariants are treated.

scholarly journals Virtual braids and virtual curve diagrams

2015 ◽  
Vol 24 (13) ◽  
pp. 1541001 ◽  
Author(s):  
Oleg Chterental
Keyword(s):  
Automorphism Group ◽  
Braid Group ◽  
Knot Theory ◽  
Class Group ◽  
Mapping Class ◽  
Virtual Knot ◽  
Injective Homomorphism ◽  
Virtual Links ◽  
Upper Half Plane ◽  
Virtual Knot Theory

There is a well-known injective homomorphism [Formula: see text] from the classical braid group [Formula: see text] into the automorphism group of the free group [Formula: see text], first described by Artin [Theory of Braids, Ann. Math. (2) 48(1) (1947) 101–126]. This homomorphism induces an action of [Formula: see text] on [Formula: see text] that can be recovered by considering the braid group as the mapping class group of [Formula: see text] (an upper half plane with [Formula: see text] punctures) acting naturally on the fundamental group of [Formula: see text]. Kauffman introduced virtual links [Virtual knot theory, European J. Combin. 20 (1999) 663–691] as an extension of the classical notion of a link in [Formula: see text]. There is a corresponding notion of a virtual braid, and the set of virtual braids on [Formula: see text] strands forms a group [Formula: see text]. In this paper, we will generalize the Artin action to virtual braids. We will define a set, [Formula: see text], of “virtual curve diagrams” and define an action of [Formula: see text] on [Formula: see text]. Then, we will show that, as in Artin’s case, the action is faithful. This provides a combinatorial solution to the word problem in [Formula: see text]. In the papers [V. G. Bardakov, Virtual and welded links and their invariants, Siberian Electron. Math. Rep. 21 (2005) 196–199; V. O. Manturov, On recognition of virtual braids, Zap. Nauch. Sem. POMI 299 (2003) 267–286], an extension [Formula: see text] of the Artin homomorphism was introduced, and the question of its injectivity was raised. We find that [Formula: see text] is not injective by exhibiting a non-trivial virtual braid in the kernel when [Formula: see text].

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