PERIODIC VIRTUAL LINKS AND THE BINARY 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.

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Cyclic coverings of virtual link diagrams

International Journal of Mathematics ◽

10.1142/s0129167x19500721 ◽

2019 ◽

Vol 30 (14) ◽

pp. 1950072 ◽

Cited By ~ 1

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.

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Recurrent Generalization of F-Polynomials for Virtual Knots and Links

Symmetry ◽

10.3390/sym14010015 ◽

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.

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Linking invariants of even virtual links

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216517500729 ◽

2017 ◽

Vol 26 (12) ◽

pp. 1750072 ◽

Cited By ~ 1

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.

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Wirtinger numbers for virtual links

Journal of Knot Theory and Its Ramifications ◽

10.1142/s021821651950086x ◽

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.

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CLASSIFICATION OF CLOSED VIRTUAL 2-BRAIDS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216508006592 ◽

2008 ◽

Vol 17 (10) ◽

pp. 1223-1239 ◽

Cited By ~ 2

Author(s):

TERUHISA KADOKAMI

Keyword(s):

Braid Group ◽

Existence And Uniqueness ◽

Minimal Realization ◽

Virtual Link ◽

Virtual Links ◽

Bracket Polynomial

We classify closed virtual 2-braids completely as virtual links. For the proof, we use surface bracket polynomial due to Dye and Kauffman, Kuperberg's theorem which states existence and uniqueness of a minimal realization for a virtual link, and a subgroup of the 3-braid group.

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Graphical virtual links and a polynomial for signed cyclic graphs

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216518500542 ◽

2018 ◽

Vol 27 (10) ◽

pp. 1850054 ◽

Cited By ~ 1

Author(s):

Qingying Deng ◽

Xian’an Jin ◽

Louis H. Kauffman

Keyword(s):

The Other ◽

Virtual Link ◽

Spanning Subgraph ◽

Link Diagrams ◽

Virtual Links ◽

Cyclic Graphs ◽

Polynomial Formula ◽

Bracket Polynomial ◽

The Relationship ◽

Cyclic Graph

For a signed cyclic graph [Formula: see text], we can construct a unique virtual link [Formula: see text] by taking the medial construction and converting 4-valent vertices of the medial graph to crossings according to the signs. If a virtual link can occur in this way then we say that the virtual link is graphical. In this paper, we shall prove that a virtual link [Formula: see text] is graphical if and only if it is checkerboard colorable. On the other hand, we introduce a polynomial [Formula: see text] for signed cyclic graphs, which is defined via a deletion-marking recursion. We shall establish the relationship between [Formula: see text] of a signed cyclic graph [Formula: see text] and the bracket polynomial of one of the virtual link diagrams associated with [Formula: see text]. Finally, we give a spanning subgraph expansion for [Formula: see text].

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

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216521500012 ◽

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.

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Similarity indices and the Miyazawa polynomials of virtual links

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216514600037 ◽

2014 ◽

Vol 23 (07) ◽

pp. 1460003 ◽

Cited By ~ 1

Author(s):

Myeong-Ju Jeong ◽

Chan-Young Park

Keyword(s):

Knot Theory ◽

State Model ◽

Necessary Condition ◽

Virtual Link ◽

Polynomial Invariant ◽

Virtual Knots ◽

Vassiliev Invariant ◽

Virtual Links ◽

Bracket Polynomial ◽

Knots And Links

Y. Miyazawa introduced a two-variable polynomial invariant of virtual knots in 2006 [Magnetic graphs and an invariant for virtual links, J. Knot Theory Ramifications 15 (2006) 1319–1334] and then generalized it to give a multi-variable one via decorated virtual magnetic graph diagrams in 2008. A. Ishii gave a simple state model for the two-variable Miyazawa polynomial by using pole diagrams in 2008 [A multi-variable polynomial invariant for virtual knots and links, J. Knot Theory Ramifications 17 (2008) 1311–1326]. H. A. Dye and L. H. Kauffman constructed an arrow polynomial of a virtual link in 2009 which is equivalent to the multi-variable Miyazawa polynomial [Virtual crossing number and the arrow polynomial, preprint (2008), arXiv:0810.3858v3, http://front.math.ucdavis.edu .]. We give a bracket model for the multi-variable Miyazawa polynomial via pole diagrams and polar tangles similarly to the Ishii's state model for the two-variable polynomial. By normalizing the bracket polynomial we get the multi-variable Miyazawa polynomial fK ∈ ℤ[A, A-1, K1, K2, …] of a virtual link K. n-similar knots take the same value for any Vassiliev invariant of degree < n. We show that fK1 ≡ fK2 mod (A4 - 1)n if two virtual links K1 and K2 are n-similar. Also we give a necessary condition for a virtual link to be periodic by using n-similarity of virtual tangles and the Miyazawa polynomial.

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Coherent double coverings of virtual link diagrams

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216518430046 ◽

2018 ◽

Vol 27 (11) ◽

pp. 1843004 ◽

Cited By ~ 1

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.

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