On Gauss diagrams of periodic virtual knots

2015 ◽  
Vol 24 (10) ◽  
pp. 1540008 ◽  
Author(s):  
Yongju Bae ◽  
In Sook Lee
Keyword(s):  
Virtual Knot ◽  
Virtual Knots ◽  
Gauss Diagram

In this paper, we study the Gauss diagrams for periodic virtual knots (Theorem 3.1) and show that the virtual knot corresponding to a periodic Gauss diagram is equivalent to the periodic virtual knot whose factor is the virtual knot corresponding to the factor Gauss diagram (Theorem 3.2). We give formulae for the writhe polynomial and the affine index polynomial of periodic virtual knots by using those of factor knots (Corollary 4.2, Corollary 4.6).

scholarly journals UNKNOTTING VIRTUAL KNOTS WITH GAUSS DIAGRAM FORBIDDEN MOVES

2001 ◽  
Vol 10 (06) ◽  
pp. 931-935 ◽  
Author(s):  
SAM NELSON
Keyword(s):  
Virtual Knot ◽  
Virtual Knots ◽  
Reidemeister Moves ◽  
Gauss Diagram

The forbidden moves can be combined with Gauss diagram Reidemeister moves to obtain move sequences with which we may change any Gauss diagram (and hence any virtual knot) into any other, including in particular the unknotted diagram.

scholarly journals PARITY AND EXOTIC COMBINATORIAL FORMULAE FOR FINITE-TYPE INVARIANTS OF VIRTUAL KNOTS

2012 ◽  
Vol 21 (13) ◽  
pp. 1240001 ◽  
Author(s):  
MICAH WHITNEY CHRISMAN ◽  
VASSILY OLEGOVICH MANTUROV
Keyword(s):  
Finite Type ◽  
Combinatorial Formula ◽  
Knot Invariants ◽  
Virtual Knot ◽  
Additional Information ◽  
Virtual Knots ◽  
Finite Type Invariants ◽  
Gauss Diagram

The present paper produces examples of Gauss diagram formulae for virtual knot invariants which have no analogue in the classical knot case. These combinatorial formulae contain additional information about how a subdiagram is embedded in a virtual knot diagram. The additional information comes from the second author's recently discovered notion of parity. For a parity of flat virtual knots, the new combinatorial formulae are Kauffman finite-type invariants. However, many of the combinatorial formulae possess exotic properties. It is shown that there exists an integer-valued virtualization invariant combinatorial formula of order n for every n (i.e. it is stable under the map which changes the direction of one arrow but preserves the sign). Hence, it is not of Goussarov–Polyak–Viro finite-type. Moreover, every homogeneous Polyak–Viro combinatorial formula admits a decomposition into an "even" part and an "odd" part. For the Gaussian parity, neither part of the formula is of GPV finite-type when it is non-constant on the set of classical knots. In addition, eleven new non-trivial combinatorial formulae of order 2 are presented which are not of GPV finite-type.

scholarly journals A note on coverings of virtual knots

2019 ◽  
Vol 29 (08) ◽  
pp. 1971002
Author(s):  
Takuji Nakamura ◽  
Yasutaka Nakanishi ◽  
Shin Satoh
Keyword(s):  
Virtual Knot ◽  
Virtual Knots ◽  
Finite Set ◽  
Gauss Diagram

For a virtual knot [Formula: see text] and an integer [Formula: see text], the [Formula: see text]-covering [Formula: see text] is defined by using the indices of chords on a Gauss diagram of [Formula: see text]. In this paper, we prove that for any finite set of virtual knots [Formula: see text], there is a virtual knot [Formula: see text] such that [Formula: see text], [Formula: see text], and otherwise [Formula: see text].

Finiteness of the set of virtual knots with a given state number

2018 ◽  
Vol 27 (08) ◽  
pp. 1850049
Author(s):  
Takuji Nakamura ◽  
Yasutaka Nakanishi ◽  
Shin Satoh
Keyword(s):  
The Real ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Minimum Number ◽  
Number Formula

A state of a virtual knot diagram [Formula: see text] is a collection of circles obtained from [Formula: see text] by splicing all the real crossings. For each integer [Formula: see text], we denote by [Formula: see text] the number of states of [Formula: see text] with [Formula: see text] circles. The [Formula: see text]-state number [Formula: see text] of a virtual knot [Formula: see text] is the minimum number of [Formula: see text] for [Formula: see text] of [Formula: see text]. Let [Formula: see text] be the set of virtual knots [Formula: see text] with [Formula: see text] for an integer [Formula: see text]. In this paper, we study the finiteness of [Formula: see text]. We determine the finiteness of [Formula: see text] for any [Formula: see text] and [Formula: see text] for any [Formula: see text].

Picture-valued parity-biquandle bracket

2020 ◽  
Vol 29 (02) ◽  
pp. 2040004 ◽  
Author(s):  
Denis P. Ilyutko ◽  
Vassily O. Manturov
Keyword(s):  
Classical Case ◽  
Linear Combinations ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Underlying Graph ◽  
Kauffman Bracket ◽  
Knot Diagrams

In V. O. Manturov, On free knots, preprint (2009), arXiv:math.GT/0901.2214], the second named author constructed the bracket invariant [Formula: see text] of virtual knots valued in pictures (linear combinations of virtual knot diagrams with some crossing information omitted), such that for many diagrams [Formula: see text], the following formula holds: [Formula: see text], where [Formula: see text] is the underlying graph of the diagram, i.e. the value of the invariant on a diagram equals the diagram itself with some crossing information omitted. This phenomenon allows one to reduce many questions about virtual knots to questions about their diagrams. In [S. Nelson, M. E. Orrison and V. Rivera, Quantum enhancements and biquandle brackets, preprint (2015), arXiv:math.GT/1508.06573], the authors discovered the following phenomenon: having a biquandle coloring of a certain knot, one can enhance various state-sum invariants (say, Kauffman bracket) by using various coefficients depending on colors. Taking into account that the parity can be treated in terms of biquandles, we bring together the two ideas from these papers and construct the picture-valued parity-biquandle bracket for classical and virtual knots. This is an invariant of virtual knots valued in pictures. Both the parity bracket and Nelson–Orrison–Rivera invariants are partial cases of this invariant, hence this invariant enjoys many properties of various kinds. Recently, the authors together with E. Horvat and S. Kim have found that the picture-valued phenomenon works in the classical case.

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.

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.

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.

On the virtual Alexander polynomial of periodic virtual knots

2014 ◽  
Vol 23 (07) ◽  
pp. 1460011 ◽  
Author(s):  
Joonoh Kim ◽  
Sang Youl Lee ◽  
Myoungsoo Seo
Keyword(s):  
Alexander Polynomial ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Period 2

In this paper, we give a relationship between the virtual Alexander polynomial of a periodic virtual knot and that of its factor knot. As an application, we improve Im–Lee's table of possible periods of 117 virtual knots with classical crossings ≤ 4 in Jeremy Green's table. In particular, we prove that the virtual knots 4.1 and 4.77 have the actual period 2 and no others, and the virtual knot 4.99 has the actual period 2.

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