THE GORDIAN COMPLEX OF VIRTUAL KNOTS

2012 ◽  
Vol 21 (14) ◽  
pp. 1250122 ◽  
Author(s):  
SUMIKO HORIUCHI ◽  
KASUMI KOMURA ◽  
YOSHIYUKI OHYAMA ◽  
MASAFUMI SHIMOZAWA
Keyword(s):  
Natural Number ◽  
Simplicial Complex ◽  
Virtual Knot ◽  
Virtual Knots ◽  
The Family

Hirasawa and Uchida defined the Gordian complex of knots which is a simplicial complex whose vertices consist of all knot types in S3. In this paper, we define the Gordian complex of virtual knots which is a simplicial complex whose vertices consist of all virtual knots by using the local move which makes a real crossing a virtual crossing. We show that for any virtual knot K0 and for any given natural number n, there exists a family of virtual knots {K0, K1,…,Kn} such that for any pair (Ki, Kj) of distinct elements of the family, the Gordian distance of virtual knots dv(Ki, Kj) = 1. And we also give a formula of the f-polynomial for the sum of tangles of virtual knots.

THE GORDIAN COMPLEX OF VIRTUAL KNOTS BY FORBIDDEN MOVES

2013 ◽  
Vol 22 (09) ◽  
pp. 1350051 ◽  
Author(s):  
SUMIKO HORIUCHI ◽  
YOSHIYUKI OHYAMA
Keyword(s):  
Natural Number ◽  
Simplicial Complex ◽  
Virtual Knot ◽  
Virtual Knots ◽  
The Family

Hirasawa and Uchida defined the Gordian complex of knots which is a simplicial complex whose vertices consist of all knot types in S3. In this paper, we define the Gordian complex of virtual knots by using forbidden moves. We show that for any virtual knot K0 and for any given natural number n, there exists a family of virtual knots {K0, K1, …, Kn} such that for any pair (Ki, Kj) of distinct elements of the family, the Gordian distance of virtual knots by forbidden moves dF(Ki, Kj) = 1.

scholarly journals Gordian complexes of knots and virtual knots given by region crossing changes and arc shift moves

2020 ◽  
Vol 29 (10) ◽  
pp. 2042008
Author(s):  
Amrendra Gill ◽  
Madeti Prabhakar ◽  
Andrei Vesnin
Keyword(s):  
Simplicial Complex ◽  
Infinite Family ◽  
High Dimensional ◽  
Dimensional Simplex ◽  
Virtual Knot ◽  
Virtual Knots ◽  
The Family

Gordian complex of knots was defined by Hirasawa and Uchida as the simplicial complex whose vertices are knot isotopy classes in [Formula: see text]. Later Horiuchi and Ohyama defined Gordian complex of virtual knots using [Formula: see text]-move and forbidden moves. In this paper, we discuss Gordian complex of knots by region crossing change and Gordian complex of virtual knots by arc shift move. Arc shift move is a local move in the virtual knot diagram which results in reversing orientation locally between two consecutive crossings. We show the existence of an arbitrarily high-dimensional simplex in both the Gordian complexes, i.e. by region crossing change and by the arc shift move. For any given knot (respectively, virtual knot) diagram we construct an infinite family of knots (respectively, virtual knots) such that any two distinct members of the family have distance one by region crossing change (respectively, arc shift move). We show that the constructed virtual knots have the same affine index polynomial.

THE Ck-GORDIAN COMPLEX OF KNOTS

2006 ◽  
Vol 15 (01) ◽  
pp. 73-80 ◽  
Author(s):  
YOSHIYUKI OHYAMA
Keyword(s):  
Natural Number ◽  
Simplicial Complex ◽  
The Family

Hirasawa and Uchida defined the Gordian complex of knots which is a simplicial complex whose vertices consist of all knot types in S3by using "a crossing change". In this paper, we define the Ck-Gordian complex of knots which is an extension of the Gordian complex of knots. Let k be a natural number more than 2 and we show that for any knot K0and any given natural number n, there exists a family of knots {K0, K1,…, Kn} such that for any pair (Ki, Kj) of distinct elements of the family, the Ck-distance dCk(Ki, Kj) = 1.

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.

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.

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 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.

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