Delta moves and Kauffman polynomials of virtual knots

2014 ◽  
Vol 23 (10) ◽  
pp. 1450053 ◽  
Author(s):  
Myeong-Ju Jeong
Keyword(s):  
Lower Bound ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Vassiliev Invariant ◽  
Kauffman Polynomial

In 1990, Okada showed that the second coefficients of the Conway polynomials of two knots differ by 1 if the two knots are related by a single Δ-move. We extend the Okada's result for virtual knots by using a Vassiliev invariant v2 of virtual knots of degree 2 which is induced from the Kauffman polynomial of a virtual knot. We show that v2(K1) - v2(K2) = ±48, if K2 is a virtual knot obtained from a virtual knot K1 by applying a Δ-move. From this we have a lower bound [Formula: see text] for the number of Δ-moves if two virtual knots K1 and K2 are related by a sequence of Δ-moves.

Clasp-pass moves and Kauffman polynomials of virtual knots

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.

A LOWER BOUND FOR THE NUMBER OF FORBIDDEN MOVES TO UNKNOT A LONG VIRTUAL KNOT

2013 ◽  
Vol 22 (06) ◽  
pp. 1350024 ◽  
Author(s):  
MYEONG-JU JEONG
Keyword(s):  
Lower Bound ◽  
Finite Type ◽  
Finite Sequence ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Finite Type Invariants ◽  
Combinatorial Representations ◽  
Congruent Modulo

Nelson and Kanenobu showed that forbidden moves unknot any virtual knot. Similarly a long virtual knot can be unknotted by a finite sequence of forbidden moves. Goussarov, Polyak and Viro introduced finite type invariants of virtual knots and long virtual knots and gave combinatorial representations of finite type invariants. We introduce Fn-moves which generalize the forbidden moves. Assume that two long virtual knots K and K′ are related by a finite sequence of Fn-moves. We show that the values of the finite type invariants of degree 2 of K and K′ are congruent modulo n and give a lower bound for the number of Fn-moves needed to transform K to K′.

scholarly journals An Orientation-Sensitive Vassiliev Invariant for Virtual Knots

2003 ◽  
Vol 12 (06) ◽  
pp. 767-779 ◽  
Author(s):  
Jörg Sawollek
Keyword(s):  
Virtual Link ◽  
Zeroth Order ◽  
Opposite Orientation ◽  
Vassiliev Invariants ◽  
Virtual Knot ◽  
First Order ◽  
Virtual Knots ◽  
Conway Polynomial ◽  
Vassiliev Invariant ◽  
Open Question

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.

AN ESTIMATE OF THE UNKNOTTING NUMBERS FOR VIRTUAL KNOTS BY FORBIDDEN MOVES

2013 ◽  
Vol 22 (03) ◽  
pp. 1350009 ◽  
Author(s):  
MIGIWA SAKURAI
Keyword(s):  
Lower Bound ◽  
Finite Sequence ◽  
Virtual Knot ◽  
Virtual Knots ◽  
The Difference

It is known that any virtual knot can be deformed into the trivial knot by a finite sequence of forbidden moves. In this paper, we give the difference of the values obtained from some invariants constructed by Henrich between two virtual knots which can be transformed into each other by a single forbidden move. As a result, we obtain a lower bound of the unknotting number of a virtual knot by forbidden moves.

Classification of genus 1 virtual knots having at most five classical crossings

2014 ◽  
Vol 23 (06) ◽  
pp. 1450031 ◽  
Author(s):  
Akimova Alena Andreevna ◽  
Sergei Vladimirovich Matveev
Keyword(s):  
Virtual Knot ◽  
Virtual Knots ◽  
Kauffman Polynomial ◽  
Thickened Torus ◽  
Knot Diagrams ◽  
Genus One

The goal of this paper is to tabulate all genus one prime virtual knots having diagrams with ≤ 5 classical crossings. First, we construct all nonlocal prime knots in the thickened torus T × I which have diagrams with ≤ 5 crossings and admit no destabilizations. Then we use a generalized version of the Kauffman polynomial to prove that all those knots are different. Finally, we convert the knot diagrams in T thus obtained into virtual knot diagrams in the plane.

A zero polynomial of virtual knots

2016 ◽  
Vol 25 (01) ◽  
pp. 1550078 ◽  
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.

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.

VASSILIEV INVARIANTS FOR VIRTUAL LINKS, CURVES ON SURFACES AND THE JONES–KAUFFMAN POLYNOMIAL

2005 ◽  
Vol 14 (02) ◽  
pp. 231-242 ◽  
Author(s):  
VASSILY O. MANTUROV
Keyword(s):  
Free Module ◽  
Equivalence Classes ◽  
Two Dimensional ◽  
Infinite Dimensional ◽  
Vassiliev Invariants ◽  
Virtual Knots ◽  
Vassiliev Invariant ◽  
Virtual Links ◽  
Kauffman Polynomial ◽  
Curves On Surfaces

We discuss the strong invariant of virtual links proposed in [23]. This invariant is obtained as a generalization of the Jones–Kauffman polynomial (generalized Kauffman's bracket) by adding to the sum some equivalence classes of curves in two-dimensional surfaces. Thus, the invariant is valued in the infinite-dimensional free module over Z[q,q-1]. We prove that this invariant can be decomposed into finite type Vassiliev invariant of virtual links (in Kauffman's sense); thus we present new infinite series of Vassiliev invariants. It is also proved that this invariant is strictly stronger than the Jones–Kauffman polynomial for virtual knots proposed by Kauffman. Some examples when the invariant can recognize virtual knots that can not be recognized by other invariants are given.

KAUFFMAN-LIKE POLYNOMIAL AND CURVES IN 2-SURFACES

2003 ◽  
Vol 12 (08) ◽  
pp. 1145-1153 ◽  
Author(s):  
VASSILY O. MANTUROV
Keyword(s):  
Infinite Dimensional ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Knot Invariant ◽  
Kauffman Polynomial ◽  
Dimensional Module

In the present paper, we construct a virtual knot invariant with values in the free infinite-dimensional module over Z[a, a-1]. The restriction of this invariant to the set of classical knots coincides with the Jones–Kauffman polynomial. It distinguishes virtual knots stronger than the generalised Jones-Kauffman polynomial proposed in [Kau].

THE PARITY WRITHE POLYNOMIALS FOR VIRTUAL KNOTS AND FLAT VIRTUAL KNOTS

2013 ◽  
Vol 22 (01) ◽  
pp. 1250133 ◽  
Author(s):  
YOUNG HO IM ◽  
SERA KIM ◽  
DONG SOO LEE
Keyword(s):  
Polynomial Invariant ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Vassiliev Invariant

We introduce a polynomial invariant with two variables for an oriented virtual knot, which refines the odd writhe polynomial with one variable due to Cheng by using a modified version of the warping degree. Our invariant is a Vassiliev invariant of degree one, reduces to one variable for a checkerboard colorable virtual knot, vanishes for classical knots, and detects non-invertibility and non-amphicheirality for some cases. We raise some examples to show effectiveness of our invariant. Moreover we define a similar invariant for a flat virtual knot.

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