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.

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.

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.

scholarly journals Commutator Length of Finitely Generated Linear Groups

2008 ◽  
Vol 2008 ◽  
pp. 1-5
Author(s):  
Mahboubeh Alizadeh Sanati
Keyword(s):  
Finite Group ◽  
Upper Bound ◽  
Quadratic Polynomial ◽  
Linear Groups ◽  
Number Of Generators ◽  
Derived Subgroup ◽  
Finite Linear Group ◽  
Minimum Number ◽  
Commutator Length ◽  
Finite Linear

The commutator length “” of a group is the least natural number such that every element of the derived subgroup of is a product of commutators. We give an upper bound for when is a -generator nilpotent-by-abelian-by-finite group. Then, we give an upper bound for the commutator length of a soluble-by-finite linear group over that depends only on and the degree of linearity. For such a group , we prove that is less than , where is the minimum number of generators of (upper) triangular subgroup of and is a quadratic polynomial in . Finally we show that if is a soluble-by-finite group of Prüffer rank then , where is a quadratic polynomial in .

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.

Conflict-Free Colourings of Uniform Hypergraphs With Few Edges

2012 ◽  
Vol 21 (4) ◽  
pp. 611-622 ◽  
Author(s):  
A. KOSTOCHKA ◽  
M. KUMBHAT ◽  
T. ŁUCZAK
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Upper Bounds ◽  
Uniform Hypergraph ◽  
Lower And Upper Bounds ◽  
Uniform Hypergraphs ◽  
Minimum Number

A colouring of the vertices of a hypergraph is called conflict-free if each edge e of contains a vertex whose colour does not repeat in e. The smallest number of colours required for such a colouring is called the conflict-free chromatic number of , and is denoted by χCF(). Pach and Tardos proved that for an (2r − 1)-uniform hypergraph with m edges, χCF() is at most of the order of rm1/r log m, for fixed r and large m. They also raised the question whether a similar upper bound holds for r-uniform hypergraphs. In this paper we show that this is not necessarily the case. Furthermore, we provide lower and upper bounds on the minimum number of edges of an r-uniform simple hypergraph that is not conflict-free k-colourable.

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.

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.

Parities and polynomial invariants for virtual links

2014 ◽  
Vol 23 (12) ◽  
pp. 1450066 ◽  
Author(s):  
Young Ho Im ◽  
Kyoung Il Park ◽  
Mi Hwa Shin
Keyword(s):  
Knot Theory ◽  
Virtual Link ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Virtual Knots ◽  
Link Diagrams ◽  
Virtual Links ◽  
Kauffman Polynomial

We introduce the odd Jones–Kauffman polynomial and odd Miyazawa polynomials of virtual link diagrams by using the parity of virtual link diagrams given in [Y. H. Im and K. I. Park, A parity and a multi-variable polynomial invariant for virtual links, J. Knot Theory Ramifications22(13) (2013), Article ID: 1350073, 18pp.], which are different from the original Jones–Kauffman and Miyazawa polynomials. Also, we give a family of parities and odd polynomials for virtual knots so that many virtual knots can be distinguished.

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.

An upper bound for the breadth of the Jones polynomial

1988 ◽  
Vol 103 (3) ◽  
pp. 451-456 ◽  
Author(s):  
Morwen B. Thistlethwaite
Keyword(s):  
Simple Proof ◽  
Upper Bound ◽  
Recent Article ◽  
Jones Polynomial ◽  
Upper Bounds ◽  
The Other ◽  
Link Diagram ◽  
Kauffman Bracket ◽  
Bracket Polynomial ◽  
Connected Diagram

In the recent article [2], a kind of connected link diagram called adequate was investigated, and it was shown that the Jones polynomial is never trivial for such a diagram. Here, on the other hand, upper bounds are considered for the breadth of the Jones polynomial of an arbitrary connected diagram, thus extending some of the results of [1,4,5]. Also, in Theorem 2 below, a characterization is given of those connected, prime diagrams for which the breadth of the Jones polynomial is one less than the number of crossings; recall from [1,4,5] that the breadth equals the number of crossings if and only if that diagram is reduced alternating. The article is concluded with a simple proof, using the Jones polynomial, of W. Menasco's theorem [3] that a connected, alternating diagram cannot represent a split link. We shall work with the Kauffman bracket polynomial 〈D〉 ∈ Z[A, A−1 of a link diagram D.

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