An Orientation-Sensitive Vassiliev Invariant for Virtual Knots

It is an open question whether there are Vassiliev invariants that can distinguish an oriented knot from its inverse, i.e., the knot with the opposite orientation. In this article, an example is given for a first order Vassiliev invariant that takes different values on a virtual knot and its inverse. The Vassiliev invariant is derived from the Conway polynomial for virtual knots. Furthermore, it is shown that the zeroth order Vassiliev invariant coming from the Conway polynomial cannot distinguish a virtual link from its inverse and that it vanishes for virtual knots.

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INVARIANTS OF FLAT VIRTUAL LINKS INDUCED FROM VASSILIEV INVARIANTS OF DEGREE ONE FOR VIRTUAL LINKS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216511009479 ◽

2011 ◽

Vol 20 (12) ◽

pp. 1649-1667 ◽

Cited By ~ 3

Author(s):

YOUNG HO IM ◽

SERA KIM ◽

KYEONGHUI LEE

Keyword(s):

Vassiliev Invariants ◽

Virtual Knot ◽

Virtual Knots ◽

Virtual Links

We introduce invariants of flat virtual links which are induced from Vassiliev invariants of degree one for virtual links. Also we give several properties of these invariants for flat virtual links and examples. In particular, if the value of some invariants of flat virtual knots F are non-zero, then F is non-invertible so that every virtual knot overlying F is non-invertible.

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LOCAL MOVES AND GORDIAN COMPLEXES

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216506005068 ◽

2006 ◽

Vol 15 (09) ◽

pp. 1215-1224 ◽

Cited By ~ 4

Author(s):

YASUTAKA NAKANISHI ◽

YOSHIYUKI OHYAMA

Keyword(s):

Vassiliev Invariants ◽

Conway Polynomial ◽

Vassiliev Invariant ◽

The Relationship

By the works of Gusarov [2] and Habiro [3], it is known that a local move called the Cnmove is strongly related to Vassiliev invariants of order less than n. The coefficient of the znterm in the Conway polynomial is known to be a Vassiliev invariant of order n. In this note, we will consider to what degree the relationship is strong with respect to Conway polynomial. Let K be a knot, and KCnthe set of knots obtained from a knot K by a single Cnmove. Let [Formula: see text] be the set of the Conway polynomials [Formula: see text] for a set of knots [Formula: see text]. Our main result is the following: There exists a pair of knots K1, K2such that ∇K1= ∇K2and [Formula: see text]. In other words, the CnGordian complex is not homogeneous with respect to Conway polynomial.

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A Cn-MOVE FOR A KNOT AND THE COEFFICIENTS OF THE CONWAY POLYNOMIAL

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216508006403 ◽

2008 ◽

Vol 17 (07) ◽

pp. 771-785 ◽

Cited By ~ 6

Author(s):

YOSHIYUKI OHYAMA ◽

HARUMI YAMADA

Keyword(s):

Vassiliev Invariants ◽

Conway Polynomial ◽

Vassiliev Invariant

It is shown that two knots can be transformed into each other by Cn-moves if and only if they have the same Vassiliev invariants of order less than n. Consequently, a Cn-move cannot change the Vassiliev invariants of order less than n and may change those of order more than or equal to n. In this paper, we consider the coefficient of the Conway polynomial as a Vassiliev invariant and show that a Cn-move changes the nth coefficient of the Conway polynomial by ±2, or 0. And for the 2mth coefficient (2m > n), it can change by p or p + 1 for any given integer p.

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ON A LOCAL MOVE FOR VIRTUAL KNOTS AND LINKS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216509007592 ◽

2009 ◽

Vol 18 (11) ◽

pp. 1577-1596 ◽

Cited By ~ 1

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).

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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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Clasp-pass moves and Kauffman polynomials of virtual knots

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216516500450 ◽

2016 ◽

Vol 25 (08) ◽

pp. 1650045

Author(s):

Myeong-Ju Jeong ◽

Dahn-Goon Kim

Keyword(s):

Lower Bound ◽

Jones Polynomial ◽

Finite Sequence ◽

Vassiliev Invariants ◽

Type Formula ◽

Virtual Knots ◽

Vassiliev Invariant ◽

Kauffman Polynomial

Habiro showed that two knots [Formula: see text] and [Formula: see text] are related by a finite sequence of clasp-pass moves, if and only if they have the same value for Vassiliev invariants of type [Formula: see text]. Tsukamoto showed that, if two knots differ by a clasp-pass move then the values of the Vassiliev invariant [Formula: see text] of degree [Formula: see text] for the two knots differ by [Formula: see text] or [Formula: see text], where [Formula: see text] is the Jones polynomial of a knot [Formula: see text]. If two virtual knots are related by clasp-pass moves, then they take the same value for all Vassiliev invariants of degree [Formula: see text]. We extend the Tsukamoto’s result to virtual knots by using a Vassiliev invariant [Formula: see text] of degree [Formula: see text], which is induced from the Kauffman polynomial. We also get a lower bound for the minimal number of clasp-pass moves needed to transform [Formula: see text] to [Formula: see text], if two virtual knots [Formula: see text] and [Formula: see text] can be related by a finite sequence of clasp-pass moves.

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TWIST MOVES AND VASSILIEV INVARIANTS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s021821650400341x ◽

2004 ◽

Vol 13 (06) ◽

pp. 719-735

Author(s):

MYEONG-JU JEONG ◽

EUN-JIN KIM ◽

CHAN-YOUNG PARK

Keyword(s):

Vassiliev Invariants ◽

Conway Polynomial ◽

Vassiliev Invariant ◽

Oriented Parallel ◽

Half Twist

The transforms of two oriented parallel strands to a k-half twist of two strands are called tk-move and [Formula: see text]-move respectively depending on the orientations of the two strands. In this paper we give criterions to detect whether a knot K can be transformed to a knot K' by t2k-moves and [Formula: see text]-moves respectively and if so, we give some results on how many moves are needed in these transformations respectively, by using some Vassiliev invariants. Moreover we give a relation between the Δ-move and the t2k-move by considering the coefficient of z2 in the Conway polynomial of a knot, which is a Vassiliev invariant of degree 2.

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A PARITY AND A MULTI-VARIABLE POLYNOMIAL INVARIANT FOR VIRTUAL LINKS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216513500739 ◽

2013 ◽

Vol 22 (13) ◽

pp. 1350073 ◽

Cited By ~ 2

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.

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An index definition of parity mappings of a virtual link diagram and Vassiliev invariants of degree one

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216514600104 ◽

2014 ◽

Vol 23 (07) ◽

pp. 1460010

Author(s):

Kyeonghui Lee ◽

Young Ho Im ◽

Sunho Lee

Keyword(s):

Necessary Conditions ◽

Virtual Link ◽

Link Diagram ◽

Vassiliev Invariants ◽

Virtual Knot ◽

Definition Of

H. Dye defined the parity mapping for a virtual knot diagram, which is a map from the set of real crossings of the diagram to ℤ. The notion generalizes the parity which is studied extensively by V. Manturov. The mapping induces the ith writhe (i ∈ ℤ\{0}) which is an invariant of the representing virtual knot. She applied the parity mapping to introduce a grade to the Henrich S-invariant for a virtual knot, and showed that the invariants are Vassiliev invariants of degree one. Following it, we define the parity mappings for a virtual link diagram, and define the similar invariants as above for a virtual link by using the parity mappings. We show that some of the invariants are Vassiliev invariants of degree one. We also checked necessary conditions for invertibility and amphicheirality via the invariants.

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A zero polynomial of virtual knots

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216515500789 ◽

2016 ◽

Vol 25 (01) ◽

pp. 1550078 ◽

Cited By ~ 6

Author(s):

Myeong-Ju Jeong

Keyword(s):

Similar Type ◽

Virtual Knot ◽

Virtual Knots ◽

Vassiliev Invariant

In 2013, Cheng and Gao introduced the writhe polynomial of virtual knots and Kauffman introduced the affine index polynomial of virtual knots. We introduce a zero polynomial of virtual knots of a similar type by considering weights of a suitable collection of crossings of a virtual knot diagram. We show that the zero polynomial gives a Vassiliev invariant of degree 1. It distinguishes a pair of virtual knots that cannot be distinguished by the affine index polynomial and the writhe polynomial.

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