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.

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].

A PARITY AND A MULTI-VARIABLE POLYNOMIAL INVARIANT FOR VIRTUAL LINKS

2013 ◽  
Vol 22 (13) ◽  
pp. 1350073 ◽  
Author(s):  
YOUNG HO IM ◽  
KYOUNG IL PARK
Keyword(s):  
Knot Theory ◽  
Virtual Link ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Link Diagrams ◽  
Virtual Links ◽  
Knot Diagrams

We introduce a parity of classical crossings of virtual link diagrams which extends the Gaussian parity of virtual knot diagrams and the odd writhe of virtual links that extends that of virtual knots introduced by Kauffman [A self-linking invariants of virtual knots, Fund. Math.184 (2004) 135–158]. Also, we introduce a multi-variable polynomial invariant for virtual links by using the parity of classical crossings, which refines the index polynomial introduced in [Index polynomial invariants of virtual links, J. Knot Theory Ramifications19(5) (2010) 709–725]. As consequences, we give some properties of our invariant, and raise some examples.

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 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.

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.

scholarly journals Coherent double coverings of virtual link diagrams

2018 ◽  
Vol 27 (11) ◽  
pp. 1843004 ◽  
Author(s):  
Naoko Kamada
Keyword(s):  
Virtual Link ◽  
Link Diagram ◽  
Link Diagrams ◽  
Virtual Links ◽  
Double Coverings

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 diagram to a normal virtual link diagram. We show that the normal virtual link diagrams obtained by this method from two equivalent virtual link diagrams are equivalent. We relate this method to some invariants of virtual links.

scholarly journals Cyclic coverings of virtual link diagrams

2019 ◽  
Vol 30 (14) ◽  
pp. 1950072 ◽  
Author(s):  
Naoko Kamada
Keyword(s):  
Virtual Link ◽  
Link Diagram ◽  
Cyclic Covering ◽  
Link Diagrams ◽  
Virtual Links ◽  
Cyclic Coverings

A virtual link diagram is called mod [Formula: see text] almost classical if it admits an Alexander numbering valued in integers modulo [Formula: see text], and a virtual link is called mod [Formula: see text] almost classical if it has a mod [Formula: see text] almost classical diagram as a representative. In this paper, we introduce a method of constructing a mod [Formula: see text] almost classical virtual link diagram from a given virtual link diagram, which we call an [Formula: see text]-fold cyclic covering diagram. The main result is that [Formula: see text]-fold cyclic covering diagrams obtained from two equivalent virtual link diagrams are equivalent. Thus, we have a well-defined map from the set of virtual links to the set of mod [Formula: see text] almost classical virtual links. Some applications are also given.

Detecting Non-Triviality of Virtual Links

2003 ◽  
Vol 12 (06) ◽  
pp. 781-803 ◽  
Author(s):  
Teruhisa Kadokami
Keyword(s):  
Equivalence Classes ◽  
Geometric Method ◽  
Virtual Link ◽  
Virtual Knot ◽  
Stable Equivalence ◽  
Link Diagrams ◽  
Virtual Links ◽  
Algebraic Invariants ◽  
One To One

J. S. Carter, S. Kamada and M. Saito showed that there is one to one correspondence between the virtual Reidemeister equivalence classes of virtual link diagrams and the stable equivalence classes of link diagrams on compact oriented surfaces. Using the result, we show how to obtain the supporting genus of a projected virtual link by a geometric method. From this result, we show that a certain virtual knot which cannot be judged to be non-trivial by known algebraic invariants is non-trivial, and we suggest to classify the equivalence classes of projected virtual links by using the supporting genus.

scholarly journals INTRODUCTION TO GRAPH-LINK THEORY

2009 ◽  
Vol 18 (06) ◽  
pp. 791-823 ◽  
Author(s):  
DENIS PETROVICH ILYUTKO ◽  
VASSILY OLEGOVICH MANTUROV
Keyword(s):  
Knot Theory ◽  
Jones Polynomial ◽  
Natural Generalization ◽  
Combinatorial Theory ◽  
Equivalence Relations ◽  
Virtual Link ◽  
Virtual Knot ◽  
Link Theory ◽  
Virtual Knot Theory ◽  
Graph Link

The present paper is an introduction to a combinatorial theory arising as a natural generalization of classical and virtual knot theory. There is a way to encode links by a class of "realizable" graphs. When passing to generic graphs with the same equivalence relations we get "graph-links". On one hand graph-links generalize the notion of virtual link, on the other hand they do not detect link mutations. We define the Jones polynomial for graph-links and prove its invariance. We also prove some a generalization of the Kauffman–Murasugi–Thistlethwaite theorem on "minimal diagrams" for graph-links.

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