scholarly journals Span of the Jones polynomials of certain v-adequate virtual links

2021 ◽  
pp. 2150001
Author(s):  
Minori Okamura ◽  
Keiichi Sakai
Keyword(s):  
Type Inequality ◽  
Virtual Link ◽  
Link Diagram ◽  
Virtual Links ◽  
Jones Polynomials

It is known that the Kauffman–Murasugi–Thislethwaite type inequality becomes an equality for any (possibly virtual) adequate link diagram. We refine this condition. As an application we obtain a criterion for virtual link diagram with exactly one virtual crossing to represent a properly virtual link.

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.

scholarly journals Recurrent Generalization of F-Polynomials for Virtual Knots and Links

Symmetry ◽  
2021 ◽  
Vol 14 (1) ◽  
pp. 15
Author(s):  
Amrendra Gill ◽  
Maxim Ivanov ◽  
Madeti Prabhakar ◽  
Andrei Vesnin
Keyword(s):  
Local Symmetry ◽  
Weight Functions ◽  
Virtual Link ◽  
Link Diagram ◽  
Polynomial Invariants ◽  
Knot Invariants ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Virtual Links ◽  
Knots And Links

F-polynomials for virtual knots were defined by Kaur, Prabhakar and Vesnin in 2018 using flat virtual knot invariants. These polynomials naturally generalize Kauffman’s affine index polynomial and use smoothing in the classical crossing of a virtual knot diagram. In this paper, we introduce weight functions for ordered orientable virtual and flat virtual links. A flat virtual link is an equivalence class of virtual links with respect to a local symmetry changing a type of classical crossing in a diagram. By considering three types of smoothing in classical crossings of a virtual link diagram and suitable weight functions, there is provided a recurrent construction for new invariants. It is demonstrated by explicit examples that newly defined polynomial invariants are stronger than F-polynomials.

Linking invariants of even virtual links

2017 ◽  
Vol 26 (12) ◽  
pp. 1750072 ◽  
Author(s):  
Haruko A. Miyazawa ◽  
Kodai Wada ◽  
Akira Yasuhara
Keyword(s):  
Lower Bound ◽  
The Other ◽  
Virtual Link ◽  
Link Diagram ◽  
Linking Number ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Classical Link ◽  
Virtual Links ◽  
The Difference

A virtual link diagram is even if the virtual crossings divide each component into an even number of arcs. The set of even virtual link diagrams is closed under classical and virtual Reidemeister moves, and it contains the set of classical link diagrams. For an even virtual link diagram, we define a certain linking invariant which is similar to the linking number. In contrast to the usual linking number, our linking invariant is not preserved under the forbidden moves. In particular, for two fused isotopic even virtual link diagrams, the difference between the linking invariants of them gives a lower bound of the minimal number of forbidden moves needed to deform one into the other. Moreover, we give an example which shows that the lower bound is best possible.

scholarly journals Wirtinger numbers for virtual links

2019 ◽  
Vol 28 (14) ◽  
pp. 1950086
Author(s):  
Puttipong Pongtanapaisan
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Virtual Link ◽  
Link Diagram ◽  
Virtual Knots ◽  
Link Group ◽  
Number Of Generators ◽  
Virtual Links ◽  
Minimum Number ◽  
Bridge Number

The Wirtinger number of a virtual link is the minimum number of generators of the link group over all meridional presentations in which every relation is an iterated Wirtinger relation arising in a diagram. We prove that the Wirtinger number of a virtual link equals its virtual bridge number. Since the Wirtinger number is algorithmically computable, it gives a more effective way to calculate an upper bound for the virtual bridge number from a virtual link diagram. As an application, we compute upper bounds for the virtual bridge numbers and the quandle counting invariants of virtual knots with 6 or fewer crossings. In particular, we found new examples of nontrivial virtual bridge number one knots, and by applying Satoh’s Tube map to these knots we can obtain nontrivial weakly superslice links.

PERIODIC VIRTUAL LINKS AND THE BINARY BRACKET POLYNOMIAL

2012 ◽  
Vol 21 (03) ◽  
pp. 1250002 ◽  
Author(s):  
MYEONG-JU JEONG ◽  
CHAN-YOUNG PARK
Keyword(s):  
Positive Integer ◽  
Slight Modification ◽  
Mirror Image ◽  
Virtual Link ◽  
Link Diagram ◽  
Virtual Links ◽  
Modulo P ◽  
Bracket Polynomial

L. H. Kauffman defined the binary bracket polynomial of a virtual link by introducing binary labelings into the states of a virtual link diagram. We use the invariant by a slight modification, and call it the modified b-polynomial. We prove that if a virtual link K has a period pl for a prime p and a positive integer l, then the modified b-polynomial Inv K (A) of K is congruent to Inv K* (A) modulo p and A4pl-1 where K* is the mirror image of K. We exhibit examples of virtual links whose periods are completely determined by the invariant.

Oriented local moves and divisibility of the Jones–Kauffman polynomial

2019 ◽  
Vol 28 (14) ◽  
pp. 1950088
Author(s):  
Paul Drube ◽  
Puttipong Pongtanapaisan
Keyword(s):  
Jones Polynomial ◽  
Necessary Condition ◽  
Virtual Link ◽  
Type Formula ◽  
Virtual Links ◽  
Special Cases ◽  
Jones Polynomials ◽  
Kauffman Polynomial

For any virtual link [Formula: see text] that may be decomposed into a pair of oriented [Formula: see text]-tangles [Formula: see text] and [Formula: see text], an oriented local move of type [Formula: see text] is a replacement of [Formula: see text] with the [Formula: see text]-tangle [Formula: see text] in a way that preserves the orientation of [Formula: see text]. After developing a general decomposition for the Jones polynomial of the virtual link [Formula: see text] in terms of various (modified) closures of [Formula: see text], we analyze the Jones polynomials of virtual links [Formula: see text] that differ via a local move of type [Formula: see text]. Succinct divisibility conditions on [Formula: see text] are derived for broad classes of local moves that include the [Formula: see text]-move and the double-[Formula: see text]-move as special cases. As a consequence of our divisibility result for the double-[Formula: see text]-move, we introduce a necessary condition for any pair of classical knots to be [Formula: see text]-equivalent.

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.

ON THE JONES POLYNOMIAL OF ADEQUATE VIRTUAL LINKS

2010 ◽  
Vol 19 (07) ◽  
pp. 961-974
Author(s):  
YONGJU BAE ◽  
HYE SOOK LEE ◽  
CHAN-YOUNG PARK
Keyword(s):  
Jones Polynomial ◽  
Crossing Number ◽  
Virtual Link ◽  
Link Diagram ◽  
Virtual Links

In this paper, we prove that an adequate virtual link diagram of an adequate virtual link has minimal real crossing number.

A MULTI-VARIABLE POLYNOMIAL INVARIANT FOR UNORIENTED VIRTUAL KNOTS AND LINKS

2009 ◽  
Vol 18 (05) ◽  
pp. 625-649 ◽  
Author(s):  
YASUYUKI MIYAZAWA
Keyword(s):  
Crossing Number ◽  
Weighted Sum ◽  
Virtual Link ◽  
Polynomial Invariant ◽  
Virtual Knots ◽  
Virtual Links ◽  
Knots And Links

We construct a multi-variable polynomial invariant Y for unoriented virtual links as a certain weighted sum of polynomials, which are derived from virtual magnetic graphs with oriented vertices, on oriented virtual links associated with a given virtual link. We show some features of the Y-polynomial including an evaluation of the virtual crossing number of a virtual link.

SOME NUMERICAL INVARIANTS OF FLAT VIRTUAL LINKS ARE ALWAYS REALIZED BY REDUCED DIAGRAMS

2006 ◽  
Vol 15 (03) ◽  
pp. 289-297 ◽  
Author(s):  
TERUHISA KADOKAMI
Keyword(s):  
Intersection Number ◽  
Finite Sequence ◽  
Closed Surface ◽  
Crossing Number ◽  
The Self ◽  
Virtual Link ◽  
Virtual Knot ◽  
Virtual Links ◽  
Geodesic Loop ◽  
Numerical Invariants

Any flat virtual link has a reduced diagram which satisfies a certain minimality, and reduced diagrams are related one another by a finite sequence of a certain Reidemeister move. The move preserves some numerical invariants of diagrams. So we can define numerical invariants for flat virtual links. One of them, the crossing number of a flat virtual knot K, coinsides with the self-intersection number of K as an essential geodesic loop on a hyperbolic closed surface. We also show an equation among these numerical invariants, basic properties by using the equation, and determine non-split flat virtual links with the crossing number up to three.

Sign in / Sign up

Export Citation Format

Share Document