Polynomial invariants via odd parities for virtual link diagrams

2017 ◽  
Vol 26 (04) ◽  
pp. 1750021 ◽  
Author(s):  
Young Ho Im ◽  
Sera Kim ◽  
Kyoung Il Park
Keyword(s):  
Virtual Link ◽  
Polynomial Invariants ◽  
Odd Index ◽  
Link Diagrams ◽  
Virtual Links ◽  
Original Index

We introduce the odd index polynomial and the odd arrow polynomial for virtual links which are different from the original index polynomial and arrow polynomial.

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.

State sum invariants for flat virtual links from the chord index

2020 ◽  
Vol 29 (05) ◽  
pp. 2050027
Author(s):  
Kyeonghui Lee ◽  
Young Ho Im ◽  
Sera Kim
Keyword(s):  
Virtual Link ◽  
Polynomial Invariants ◽  
Link Diagrams ◽  
Virtual Links ◽  
Kauffman Polynomial

We introduce some polynomial invariants for flat virtual links which are similar to the Jones–Kauffman polynomial, the Miyazawa polynomial and the arrow polynomial for virtual link diagrams, and we give several properties and examples.

Some polynomial invariants of virtual links via a parity

2017 ◽  
Vol 26 (01) ◽  
pp. 1750009
Author(s):  
Young Ho Im ◽  
Sera Kim ◽  
Kyoung Il Park
Keyword(s):  
Simple Computation ◽  
Virtual Link ◽  
Polynomial Invariants ◽  
Link Diagrams ◽  
Virtual Links

We introduce the even index polynomial of virtual link diagrams and some even state sum polynomials which are the even Jones–Kauffman, even Miyazawa and even arrow polynomials. By simple computation, we can distinguish virtual links with these even polynomial invariants. Also, we give examples so that these invariants are different from the original ones.

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 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.

VIRTUAL HOMOTOPY

2010 ◽  
Vol 19 (07) ◽  
pp. 935-960 ◽  
Author(s):  
H. A. DYE ◽  
LOUIS H. KAUFFMAN
Keyword(s):  
The Other ◽  
Virtual Link ◽  
Magnus Expansion ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Virtual Links ◽  
Skein Relation

Two welded (respectively virtual) link diagrams are homotopic if one may be transformed into the other by a sequence of extended Reidemeister moves, classical Reidemeister moves, and self crossing changes. In this paper, we extend Milnor's μ and [Formula: see text] invariants to welded and virtual links. We conclude this paper with several examples, and compute the μ invariants using the Magnus expansion and Polyak's skein relation for the μ invariants.

ALEXANDER GROUPS AND VIRTUAL LINKS

2001 ◽  
Vol 10 (01) ◽  
pp. 151-160 ◽  
Author(s):  
DANIEL S. SILVER ◽  
SUSAN G. WILLIAMS
Keyword(s):  
Alexander Polynomial ◽  
Virtual Link ◽  
Polynomial Invariants ◽  
Classical Link ◽  
Virtual Links

The extended Alexander group of an oriented virtual link l of d components is defined. From its abelianization a sequence of polynomial invariants Δi (u1,…,ud, v), i=0, 1,…, is obtained. When l is a classical link, Δi reduces to the well-known ith Alexander polynomial of the link in the d variables u1v,…,udv; in particular, Δ0 vanishes.

Sign in / Sign up

Export Citation Format

Share Document