scholarly journals The Wriggle polynomial for virtual tangles

2019 ◽  
Vol 28 (14) ◽  
pp. 1950087
Author(s):  
Nicolas Petit
Keyword(s):  
The Self ◽  
Connected Sum ◽  
Vassiliev Invariants ◽  
Virtual Knots

We generalize the Wriggle polynomial, first introduced by L. Folwaczny and L. Kauffman, to the case of virtual tangles. This generalization naturally arises when considering the self-crossings of the tangle. We prove that the generalizations (and, by corollary, the original polynomial) are Vassiliev invariants of order one for virtual knots, and study some simple properties related to the connected sum of tangles.

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.

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

2011 ◽  
Vol 20 (12) ◽  
pp. 1649-1667 ◽  
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.

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.

Connected sum of virtual knots and F-polynomials

2021 ◽  
Author(s):  
Maxim Ivanov
Keyword(s):  
Infinite Family ◽  
Connected Sum ◽  
Connected Sums ◽  
Virtual Knots ◽  
Knot Diagrams

It is known that connected sum of two virtual knots is not uniquely determined and depends on knot diagrams and choosing the points to be connected. But different connected sums of the same virtual knots cannot be distinguished by Kauffman’s affine index polynomial. For any pair of virtual knots [Formula: see text] and [Formula: see text] with [Formula: see text]-dwrithe [Formula: see text] we construct an infinite family of different connected sums of [Formula: see text] and [Formula: see text] which can be distinguished by [Formula: see text]-polynomials.

scholarly journals The number of framings of a knot in a 3-manifold

2014 ◽  
Vol 23 (13) ◽  
pp. 1450072 ◽  
Author(s):  
Patricia Cahn ◽  
Vladimir Chernov ◽  
Rustam Sadykov
Keyword(s):  
Normal Bundle ◽  
Closed Curve ◽  
The Self ◽  
Connected Sum ◽  
Orientable Manifold ◽  
Intersection Points

In view of the self-linking invariant, the number |K| of framed knots in S3 with given underlying knot K is infinite. In fact, the second author previously defined affine self-linking invariants and used them to show that |K| is infinite for every knot in an orientable manifold unless the manifold contains a connected sum factor of S1 × S2; the knot K need not be zero-homologous and the manifold is not required to be compact. We show that when M is orientable, the number |K| is infinite unless K intersects a nonseparating sphere at exactly one point, in which case |K| = 2; the existence of a nonseparating sphere implies that M contains a connected sum factor of S1 × S2. For knots in nonorientable manifolds we show that if |K| is finite, then K is disorienting, or there is an orientation-preserving isotopy of the knot to itself which changes the orientation of its normal bundle, or it intersects some embedded S2 or ℝP2 at exactly one point, or it intersects some embedded S2 at exactly two points in such a way that a closed curve consisting of an arc in K between the intersection points and an arc in S2 is disorienting.

scholarly journals VASSILIEV INVARIANTS FROM PARITY MAPPINGS

2013 ◽  
Vol 22 (04) ◽  
pp. 1340008 ◽  
Author(s):  
H. A. DYE
Keyword(s):  
Vassiliev Invariants ◽  
Virtual Knots ◽  
Gauss Diagram

Parity mappings (weights) from the chords of a Gauss diagram to the integers are defined. The parity of the chords is used to construct families of invariants of Gauss diagrams and consequently, virtual knots. Each family forms a set of degree n Vassiliev invariants for n ≥ 1.

FRAMED KNOTS IN 3-MANIFOLDS AND AFFINE SELF-LINKING NUMBERS

2005 ◽  
Vol 14 (06) ◽  
pp. 791-818 ◽  
Author(s):  
VLADIMIR CHERNOV TCHERNOV
Keyword(s):  
The Self ◽  
Compact Manifolds ◽  
Not Given ◽  
Connected Sum ◽  
Linking Number ◽  
Fundamental Fact ◽  
Linking Numbers

The number |K| of non-isotopic framed knots that correspond to a given unframed knot K ⊂ S3 is infinite. This follows from the existence of the self-linking number slk of a zero homologous framed knot. We use the approach of Vassiliev–Goussarov invariants to construct "affine self-linking numbers" that are extensions of slk to the case of nonzero homologous framed knots in 3-manifolds. As a corollary we get that |K| = ∞ for all knots in an oriented (not necessarily compact) 3-manifold M that is not realizable as a connected sum (S1 × S2)# M′. This result for compact manifolds was first stated by Hoste and Przytycki. They referred to the works of McCullough for the idea of the proof, however to the best of our knowledge prior to this work the proof of this fundamental fact was not given in literature or in a preprint form. Our proof is based on different ideas. For M = (S1 × S2)# M′ we construct K in M such that |K| = 2 ≠ ∞.

scholarly journals Index polynomials for virtual tangles

2018 ◽  
Vol 27 (12) ◽  
pp. 1850073 ◽  
Author(s):  
Nicolas Petit
Keyword(s):  
Knot Theory ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Vassiliev Invariants ◽  
Basic Properties ◽  
Virtual Knots

We generalize the index polynomial invariant, originally introduced by Turaev [Cobordism of knots on surfaces, J. Topol. 1(2) (2008) 285–305] and Henrich [A sequence of degree one vassiliev invariants for virtual knots, J. Knot Theory Ramifications 19(4) (2010) 461–487], to the case of virtual tangles. Three polynomial invariants result from this generalization; we give a brief overview of their definition and some basic properties.

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].

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.

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