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.

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.

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.

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.

ON EXTREME COEFFICIENTS OF THE JONES–KAUFFMAN POLYNOMIAL FOR VIRTUAL LINKS

2006 ◽  
Vol 15 (07) ◽  
pp. 853-868
Author(s):  
ROMAN S. AVDEEV
Keyword(s):  
Knot Theory ◽  
Classical Case ◽  
Coefficient Function ◽  
Planar Curves ◽  
Vassiliev Invariants ◽  
Virtual Knots ◽  
Chord Diagrams ◽  
Virtual Links ◽  
Kauffman Polynomial

An important problem of knot theory is to find or estimate the extreme coefficients of the Jones–Kauffman polynomial for (virtual) links with a given number of classical crossings. This problem has been studied by Morton and Bae [1] and Manchón [11] for the case of classical links. It turns out that the general case can be reduced to the case when the extreme coefficient function is expressible in terms of chord diagrams (previous authors consider only d-diagrams which correspond to the classical case [9]). We find the maximal absolute values for generic chord diagrams, thus, for generic virtual knots. Also we consider the "next" coefficient of the Jones–Kauffman polynomial in terms of framed chord diagrams and find its maximal value for a given number of chords. These two functions on chord diagrams are of their own interest because there are related to the Vassiliev invariants of classical knots and J-invariants of planar curves, as mentioned in [10].

Oriented local moves and divisibility of the Jones–Kauffman polynomial

2019 ◽  
Vol 28 (14) ◽  
pp. 1950088
Author(s):  
Paul Drube ◽  
Puttipong Pongtanapaisan
Keyword(s):  
Jones Polynomial ◽  
Necessary Condition ◽  
Virtual Link ◽  
Type Formula ◽  
Virtual Links ◽  
Special Cases ◽  
Jones Polynomials ◽  
Kauffman Polynomial

For any virtual link [Formula: see text] that may be decomposed into a pair of oriented [Formula: see text]-tangles [Formula: see text] and [Formula: see text], an oriented local move of type [Formula: see text] is a replacement of [Formula: see text] with the [Formula: see text]-tangle [Formula: see text] in a way that preserves the orientation of [Formula: see text]. After developing a general decomposition for the Jones polynomial of the virtual link [Formula: see text] in terms of various (modified) closures of [Formula: see text], we analyze the Jones polynomials of virtual links [Formula: see text] that differ via a local move of type [Formula: see text]. Succinct divisibility conditions on [Formula: see text] are derived for broad classes of local moves that include the [Formula: see text]-move and the double-[Formula: see text]-move as special cases. As a consequence of our divisibility result for the double-[Formula: see text]-move, we introduce a necessary condition for any pair of classical knots to be [Formula: see text]-equivalent.

scholarly journals A SEQUENCE OF DEGREE ONE VASSILIEV INVARIANTS FOR VIRTUAL KNOTS

2010 ◽  
Vol 19 (04) ◽  
pp. 461-487 ◽  
Author(s):  
ALLISON HENRICH
Keyword(s):  
Double Point ◽  
Infinite Dimensional ◽  
Vassiliev Invariants ◽  
Virtual Knots ◽  
Vassiliev Invariant ◽  
Matrix Invariant ◽  
Virtual Strings

For ordinary knots in R3, there are no degree one Vassiliev invariants. For virtual knots, however, the space of degree one Vassiliev invariants is infinite-dimensional. We introduce a sequence of three degree one Vassiliev invariants of virtual knots of increasing strength. We demonstrate that the strongest invariant is a universal Vassiliev invariant of degree one for virtual knots in the sense that any other degree one Vassiliev invariant can be recovered from it by a certain natural construction. To prove these results, we extend the based matrix invariant introduced by Turaev for virtual strings to the class of singular flat virtual knots with one double-point.

ON HABIRO'S Cn–MOVES AND VASSILIEV INVARIANTS OF ORDER n

1999 ◽  
Vol 08 (01) ◽  
pp. 15-26 ◽  
Author(s):  
YOSHIYUKI OHYAMA ◽  
TATSUYA TSUKAMOTO
Keyword(s):  
Finite Sequence ◽  
Vassiliev Invariants ◽  
Vassiliev Invariant ◽  
Tree Diagram ◽  
The Difference

Recently it has been proved by Habiro that two knots K1 and K2 have the same Vassiliev invariants of order less than or equal to n if and only if K1 and K2 can be transformed into each other by a finite sequence of Cn+1–moves. In this paper, we show that the difference of the Vassiliev invariants of order n between two knots that can be transformed into each other by a Cn–move is equal to the value of the Vassiliev invariant for a one-branch tree diagram of order n.

SATANIC AND THELEMIC CIRCLES ON KNOTS

2013 ◽  
Vol 22 (05) ◽  
pp. 1350017 ◽  
Author(s):  
G. FLOWERS
Keyword(s):  
Geometric Interpretation ◽  
Second Order ◽  
Geometric Properties ◽  
Vassiliev Invariants ◽  
Vassiliev Invariant ◽  
Knot Diagrams

While Vassiliev invariants have proved to be a useful tool in the classification of knots, they are frequently defined through knot diagrams, and fail to illuminate any significant geometric properties the knots themselves may possess. Here, we provide a geometric interpretation of the second-order Vassiliev invariant by examining five-point cocircularities of knots, extending some of the results obtained in [R. Budney, J. Conant, K. P. Scannell and D. Sinha, New perspectives on self-linking, Adv. Math. 191(1) (2005) 78–113]. Additionally, an analysis on the behavior of other cocircularities on knots is given.

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