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.

The generalized Alexander polynomial of periodic virtual links

2019 ◽  
Vol 28 (06) ◽  
pp. 1950042
Author(s):  
Joonoh Kim ◽  
Kyoung-Tark Kim ◽  
Mi Hwa Shin
Keyword(s):  
Alexander Polynomial ◽  
State Model ◽  
Quantum Algebras ◽  
Virtual Link ◽  
Linking Numbers ◽  
Virtual Links ◽  
Polynomial Formula

In this paper, we give several simple criteria to detect possible periods and linking numbers for a given virtual link. We investigate the behavior of the generalized Alexander polynomial [Formula: see text] of a periodic virtual link [Formula: see text] via its Yang–Baxter state model given in [L. H. Kauffman and D. E. Radford, Bi-oriented quantum algebras and a generalized Alexander polynomial for virtual links, in Diagrammatic Morphisms and Applications, Contemp. Math. 318 (2003) 113–140, arXiv:math/0112280v2 [math.GT] 31 Dec 2001].

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.

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.

A NOTE ON THE CROSSING NUMBER AND THE BRAID INDEX FOR VIRTUAL LINKS

2010 ◽  
Vol 19 (07) ◽  
pp. 867-880
Author(s):  
YASUSHI TAKEDA
Keyword(s):  
Crossing Number ◽  
Virtual Link ◽  
Braid Index ◽  
Classical Link ◽  
Virtual Links

It is well known that any virtual link is described as the closure of a virtual braid. Therefore, we can define the virtual braid index. Ohyama proved an inequality for the crossing number and the braid index of a classical link. In this paper, we prove an analogous inequality for the (total) crossing number and the braid index of a virtual link.

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 The Alexander polynomial for virtual twist knots

2017 ◽  
Vol 26 (01) ◽  
pp. 1750007
Author(s):  
Isaac Benioff ◽  
Blake Mellor
Keyword(s):  
Knot Theory ◽  
Recursive Formula ◽  
Alexander Polynomial ◽  
Polynomial Invariants ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Virtual Links ◽  
Polynomial Formula

We define a family of virtual knots generalizing the classical twist knots. We develop a recursive formula for the Alexander polynomial [Formula: see text] (as defined by Silver and Williams [Polynomial invariants of virtual links, J. Knot Theory Ramifications 12 (2003) 987–1000]) of these virtual twist knots. These results are applied to provide evidence for a conjecture that the odd writhe of a virtual knot can be obtained from [Formula: see text].

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.

scholarly journals Alexander and writhe polynomials for virtual knots

2016 ◽  
Vol 25 (08) ◽  
pp. 1650050 ◽  
Author(s):  
Blake Mellor
Keyword(s):  
Knot Theory ◽  
Alexander Polynomial ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Virtual Links ◽  
New Interpretation ◽  
Polynomial Formula ◽  
Knots And Links

We give a new interpretation of the Alexander polynomial [Formula: see text] for virtual knots due to Sawollek [On Alexander–Conway polynomials for virtual knots and Links, preprint (2001), arXiv:math/9912173] and Silver and Williams [Polynomial invariants of virtual links, J. Knot Theory Ramifications 12 (2003) 987–1000], and use it to show that, for any virtual knot, [Formula: see text] determines the writhe polynomial of Cheng and Gao [A polynomial invariant of virtual links, J. Knot Theory Ramifications 22(12) (2013), Article ID: 1341002, 33pp.] (equivalently, Kauffman’s affine index polynomial [An affine index polynomial invariant of virtual knots, J. Knot Theory Ramifications 22(4) (2013), Article ID: 1340007, 30pp.]). We also use it to define a second-order writhe polynomial, and give some applications.

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