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.

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′.

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.

Geodesic graphs for a homotopy class of virtual knots

2017 ◽  
Vol 26 (13) ◽  
pp. 1750090
Author(s):  
Sumiko Horiuchi ◽  
Yoshiyuki Ohyama
Keyword(s):  
Homotopy Class ◽  
Finite Sequence ◽  
Dimensional Lattice ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Minimum Number ◽  
Vertex Set ◽  
Number Formula ◽  
Lattice Graph ◽  
Knot Diagrams

We consider a local move, denoted by [Formula: see text], on knot diagrams or virtual knot diagrams.If two (virtual) knots [Formula: see text] and [Formula: see text] are transformed into each other by a finite sequence of [Formula: see text] moves, the [Formula: see text] distance between [Formula: see text] and [Formula: see text] is the minimum number of times of [Formula: see text] moves needed to transform [Formula: see text] into [Formula: see text]. By [Formula: see text], we denote the set of all (virtual) knots which can be transformed into a (virtual) knot [Formula: see text] by [Formula: see text] moves. A geodesic graph for [Formula: see text] is the graph which satisfies the following: The vertex set consists of (virtual) knots in [Formula: see text] and for any two vertices [Formula: see text] and [Formula: see text], the distance on the graph from [Formula: see text] to [Formula: see text] coincides with the [Formula: see text] distance between [Formula: see text] and [Formula: see text]. When we consider virtual knots and a crossing change as a local move [Formula: see text], we show that the [Formula: see text]-dimensional lattice graph for any given natural number [Formula: see text] and any tree are geodesic graphs for [Formula: see text].

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.

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].

scholarly journals An improved lower bound related to the Furstenberg-Sárközy theorem

10.37236/4656 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Mark Lewko
Keyword(s):  
Lower Bound ◽  
The Other ◽  
The Difference

Let $D(n)$ denote the cardinality of the largest subset of the set $\{1,2,\ldots,n\}$ such that the difference of no pair of distinct elements is a square. A well-known theorem of Furstenberg and Sárközy states that $D(n)=o(n)$. In the other direction, Ruzsa has proven that $D(n) \gtrsim n^{\gamma}$ for $\gamma = \frac{1}{2}\left( 1 + \frac{\log 7}{\log 65} \right) \approx 0.733077$. We improve this to $\gamma = \frac{1}{2}\left( 1 + \frac{\log 12}{\log 205} \right)  \approx 0.733412$.

SOME NUMERICAL INVARIANTS OF FLAT VIRTUAL LINKS ARE ALWAYS REALIZED BY REDUCED DIAGRAMS

2006 ◽  
Vol 15 (03) ◽  
pp. 289-297 ◽  
Author(s):  
TERUHISA KADOKAMI
Keyword(s):  
Intersection Number ◽  
Finite Sequence ◽  
Closed Surface ◽  
Crossing Number ◽  
The Self ◽  
Virtual Link ◽  
Virtual Knot ◽  
Virtual Links ◽  
Geodesic Loop ◽  
Numerical Invariants

Any flat virtual link has a reduced diagram which satisfies a certain minimality, and reduced diagrams are related one another by a finite sequence of a certain Reidemeister move. The move preserves some numerical invariants of diagrams. So we can define numerical invariants for flat virtual links. One of them, the crossing number of a flat virtual knot K, coinsides with the self-intersection number of K as an essential geodesic loop on a hyperbolic closed surface. We also show an equation among these numerical invariants, basic properties by using the equation, and determine non-split flat virtual links with the crossing number up to three.

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.

scholarly journals A lower bound for a remainder term associated with the sum of digits function

1991 ◽  
Vol 34 (1) ◽  
pp. 121-142 ◽  
Author(s):  
D. M. E. Foster
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Upper Bound ◽  
Remainder Term ◽  
Fixed Integer ◽  
Sum Of Digits ◽  
Sum Of Digits Function ◽  
The Difference

For a fixed integer q≧2, every positive integer k = Σr≧0ar(q, k)qr where each ar(q, k)∈{0,1,2,…, q−1}. The sum of digits function α(q, k) Σr≧0ar(q, k) behaves rather erratically but on averaging has a uniform behaviour. In particular if , where n>1, then it is well known that A(q, n)∼½((q − 1)/log q)n logn as n → ∞. For odd values of q, a lower bound is now obtained for the difference 2S(q, n) = A(q, n)−½(q − 1))[log n/log q, where [log n/log q] denotes the greatest integer ≦log n /log q. This complements an upper bound already found.

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.

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