scholarly journals AN EXTENDED BRACKET POLYNOMIAL FOR VIRTUAL KNOTS AND LINKS

2009 ◽  
Vol 18 (10) ◽  
pp. 1369-1422 ◽  
Author(s):  
LOUIS H. KAUFFMAN
Keyword(s):  
Finite Number ◽  
Crossing Number ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Polynomial Coefficients ◽  
Bracket Polynomial ◽  
Knots And Links

This paper defines a new invariant of virtual knots and flat virtual knots. We study this invariant in two forms: the extended bracket invariant and the arrow polyomial. The extended bracket polynomial takes the form of a sum of virtual graphs with polynomial coefficients. The arrow polynomial is a polynomial with a finite number of variables for any given virtual knot or link. We show how the extended bracket polynomial can be used to detect non-classicality and to estimate virtual crossing number and genus for virtual knots and links.

A MULTI-VARIABLE POLYNOMIAL INVARIANT FOR VIRTUAL KNOTS AND LINKS

2008 ◽  
Vol 17 (11) ◽  
pp. 1311-1326 ◽  
Author(s):  
YASUYUKI MIYAZAWA
Keyword(s):  
Crossing Number ◽  
Polynomial Invariant ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Kauffman Polynomial ◽  
Knots And Links

We construct a multi-variable polynomial invariant for virtual knots and links via the concept of a decorated virtual magnetic graph diagram. The invariant is a generalization of the Jones–Kauffman polynomial for virtual knots and links. We show some features of the invariant including an evaluation of the virtual crossing number of a virtual knot or link.

A VIRTUAL LINK POLYNOMIAL AND THE VIRTUAL CROSSING NUMBER

2009 ◽  
Vol 18 (05) ◽  
pp. 605-623 ◽  
Author(s):  
YASUYUKI MIYAZAWA
Keyword(s):  
Crossing Number ◽  
Virtual Link ◽  
Polynomial Invariant ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Multiple Variables ◽  
Knots And Links ◽  
Weight Map

We construct a multi-variable polynomial invariant for virtual knots and links by using the concepts of a decorated virtual magnetic graph diagram and a weight map. We show that the invariant is a variety of virtual link polynomial with multiple variables introduced in [16] by the author and it gives a sharpened evaluation of the virtual crossing number of a virtual knot or link.

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.

ON A LOCAL MOVE FOR VIRTUAL KNOTS AND LINKS

2009 ◽  
Vol 18 (11) ◽  
pp. 1577-1596 ◽  
Author(s):  
TOSHIYUKI OIKAWA
Keyword(s):  
Sufficient Conditions ◽  
Virtual Link ◽  
Link Invariant ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Linking Number ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Necessary And Sufficient ◽  
Knots And Links

We define a local move called a CF-move on virtual link diagrams, and show that any virtual knot can be deformed into a trivial knot by using generalized Reidemeister moves and CF-moves. Moreover, we define a new virtual link invariant n(L) for a virtual 2-component link L whose virtual linking number is an integer. Then we give necessary and sufficient conditions for two virtual 2-component links to be deformed into each other by using generalized Reidemeister moves and CF-moves in terms of a virtual linking number and n(L).

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.

scholarly journals Some remarks on the chord index

2020 ◽  
Vol 29 (10) ◽  
pp. 2042003
Author(s):  
Zhiyun Cheng ◽  
Hongzhu Gao ◽  
Mengjian Xu
Keyword(s):  
General Construction ◽  
Polynomial Invariants ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Crossing Point ◽  
Knots And Links

In this paper, we discuss how to define a chord index via smoothing a real crossing point of a virtual knot diagram. Several polynomial invariants of virtual knots and links can be recovered from this general construction. We also explain how to extend this construction from virtual knots to flat virtual knots.

scholarly journals Khovanov homology, Lee homology and a Rasmussen invariant for virtual knots

2017 ◽  
Vol 26 (03) ◽  
pp. 1741001 ◽  
Author(s):  
Heather A. Dye ◽  
Aaron Kaestner ◽  
Louis H. Kauffman
Keyword(s):  
Large Class ◽  
Jones Polynomial ◽  
Khovanov Homology ◽  
Frobenius Algebra ◽  
Conjugation Operator ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Link Diagrams ◽  
Knots And Links

The paper contains an essentially self-contained treatment of Khovanov homology, Khovanov–Lee homology as well as the Rasmussen invariant for virtual knots and virtual knot cobordisms which directly applies as well to classical knots and classical knot cobordisms. We give an alternate formulation for the Manturov definition [34] of Khovanov homology [25], [26] for virtual knots and links with arbitrary coefficients. This approach uses cut loci on the knot diagram to induce a conjugation operator in the Frobenius algebra. We use this to show that a large class of virtual knots with unit Jones polynomial is non-classical, proving a conjecture in [20] and [10]. We then discuss the implications of the maps induced in the aforementioned theory to the universal Frobenius algebra [27] for virtual knots. Next we show how one can apply the Karoubi envelope approach of Bar-Natan and Morrison [3] on abstract link diagrams [17] with cross cuts to construct the canonical generators of the Khovanov–Lee homology [30]. Using these canonical generators we derive a generalization of the Rasmussen invariant [39] for virtual knot cobordisms and generalize Rasmussen’s result on the slice genus for positive knots to the case of positive virtual knots. It should also be noted that this generalization of the Rasmussen invariant provides an easy to compute obstruction to knot cobordisms in [Formula: see text] in the sense of Turaev [42].

scholarly journals Quadruple crossing number of knots and links

2013 ◽  
Vol 156 (2) ◽  
pp. 241-253 ◽  
Author(s):  
COLIN ADAMS
Keyword(s):  
Crossing Number ◽  
Bracket Polynomial ◽  
Knots And Links

AbstractA quadruple crossing is a crossing in a projection of a knot or link that has four strands of the knot passing straight through it. A quadruple crossing projection is a projection such that all of the crossings are quadruple crossings. In a previous paper, it was proved that every knot and link has a quadruple crossing projection and hence, every knot has a minimal quadruple crossing number c4(K). In this paper, we investigate quadruple crossing number, and in particular, use the span of the bracket polynomial to determine quadruple crossing number for a variety of knots and links.

scholarly journals VIRTUAL CROSSING NUMBERS FOR VIRTUAL KNOTS

2012 ◽  
Vol 21 (13) ◽  
pp. 1240009
Author(s):  
VASSILY OLEGOVICH MANTUROV
Keyword(s):  
Crossing Number ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Crossing Numbers

The aim of the present paper is to prove that the minimal number of virtual crossings for some families of virtual knots grows quadratically with respect to the minimal number of classical crossings. All previously known estimates for virtual crossing number ([D. M. Afanasiev, Refining the invariants of virtual knots by using parity, Sb. Math.201(6) (2010) 3–18; H. A. Dye and L. H. Kauffman, Virtual crossing numbers and the arrow polynomial, in The Mathematics of Knots. Theory and Applications, eds. M. Banagl and D. Vogel (Springer-Verlag, 2010); S. Satoh and Y. Tomiyama, On the crossing number of a virtual knot, Proc. Amer. Math. Soc.140 (2012) 367–376] and so on) were principally no more than linear in the number of classical crossings (or, what is the same, in the number of edges of a virtual knot diagram) and no virtual knot was found with virtual crossing number greater than the classical crossing number.

Virtual tribrackets

2019 ◽  
Vol 28 (04) ◽  
pp. 1950026
Author(s):  
Sam Nelson ◽  
Shane Pico
Keyword(s):  
Algebraic Structure ◽  
Link Diagram ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Knots And Links

We introduce virtual tribrackets, an algebraic structure for coloring regions in the planar complement of an oriented virtual knot or link diagram. We use these structures to define counting invariants of virtual knots and links and provide examples of the computation of the invariant; in particular, we show that the invariant can distinguish certain virtual knots.

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