Reidemeister moves and parity polynomials of virtual knot diagrams

2017 ◽  
Vol 26 (10) ◽  
pp. 1750051
Author(s):  
Myeong-Ju Jeong
Keyword(s):  
Lower Bounds ◽  
Virtual Knot ◽  
The Third ◽  
Reidemeister Moves ◽  
Polynomial Formula ◽  
Knot Diagrams

When two virtual knot diagrams are virtually isotopic, there is a sequence of Reidemeister moves and virtual moves relating them. I introduced a polynomial [Formula: see text] of a virtual knot diagram [Formula: see text] and gave lower bounds for the number of Reidemeister moves in deformation of virtually isotopic knot diagrams by using [Formula: see text]. In this paper, I introduce bridge diagrams and polynomials of virtual knot diagrams based on parity of crossings, and show that the polynomials give lower bounds for the number of the third Reidemeister moves. I give an example which shows that the result is distinguished from that obtained from [Formula: see text].

Reidemeister moves and a polynomial of virtual knot diagrams

2015 ◽  
Vol 24 (02) ◽  
pp. 1550010 ◽  
Author(s):  
Myeong-Ju Jeong
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Virtual Knot ◽  
Reidemeister Moves ◽  
Knot Diagrams

In 2006 C. Hayashi gave a lower bound for the number of Reidemeister moves in deformation of two equivalent knot diagrams by using writhe and cowrithe. It can be naturally extended for two virtually isotopic virtual knot diagrams. We introduce a polynomial qK(t) of a virtual knot diagram K and give lower bounds for the number of Reidemeister moves in deformation of two virtually isotopic knots by using qK(t). We give an example which shows that the polynomial qK(t) is useful to map out a sequence of Reidemeister moves to deform a virtual knot diagram to another virtually isotopic one.

scholarly journals MINIMAL UNKNOTTING SEQUENCES OF REIDEMEISTER MOVES CONTAINING UNMATCHED RII MOVES

2012 ◽  
Vol 21 (10) ◽  
pp. 1250099 ◽  
Author(s):  
CHUICHIRO HAYASHI ◽  
MIWA HAYASHI ◽  
MINORI SAWADA ◽  
SAYAKA YAMADA
Keyword(s):  
Lower Bounds ◽  
Precise Estimate ◽  
Planar Curves ◽  
Reidemeister Moves ◽  
Minimum Number ◽  
Knot Diagrams

Arnold introduced invariants J+, J- and St for generic planar curves. It is known that both J+/2 + St and J-/2 + St are invariants for generic spherical curves. Applying these invariants to underlying curves of knot diagrams, we can obtain lower bounds for the number of Reidemeister moves required for unknotting. J- /2 + St works well to count the minimum number of unmatched RII moves. However, it works only up to a factor of two for RI moves. Let w denote the writhe for a knot diagram. We show that J-/2 + St ± w/2 also gives sharp counts for the number of required RI moves, and demonstrate that it gives a precise estimate for a certain family of diagrams of the unknot with the underlying curve r = 2 + cos (nθ/(n + 1)), (0 ≤ θ ≤ 2(n + 1)π).

scholarly journals VIRTUAL KNOT DIAGRAMS AND THE WITTEN–RESHETIKHIN–TURAEV INVARIANT

2005 ◽  
Vol 14 (08) ◽  
pp. 1045-1075 ◽  
Author(s):  
H. A. DYE ◽  
LOUIS H. KAUFFMAN
Keyword(s):  
Fundamental Group ◽  
Virtual Link ◽  
Virtual Knot ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Classical Link ◽  
Kirby Calculus ◽  
Knot Diagrams

The Witten–Reshetikhin–Turaev invariant of classical link diagrams is generalized to virtual link diagrams. This invariant is unchanged by the framed Reidemeister moves and the Kirby calculus. As a result, it is also an invariant of the 3-manifolds represented by the classical link diagrams. This generalization is used to demonstrate that there are virtual knot diagrams with a non-trivial Witten–Reshetikhin–Turaev invariant and trivial 3-manifold fundamental group.

scholarly journals Up–down colorings of virtual-link diagrams and the necessity of Reidemeister moves of type II

2017 ◽  
Vol 26 (12) ◽  
pp. 1750073 ◽  
Author(s):  
Kanako Oshiro ◽  
Ayaka Shimizu ◽  
Yoshiro Yaguchi
Keyword(s):  
Type Ii ◽  
Virtual Link ◽  
Link Diagram ◽  
Virtual Knot ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Classical Link ◽  
Minimum Number ◽  
Knot Diagrams ◽  
Text Component

We introduce an up–down coloring of a virtual-link (or classical-link) diagram. The colorabilities give a lower bound of the minimum number of Reidemeister moves of type II which are needed between two [Formula: see text]-component virtual-link (or classical-link) diagrams. By using the notion of a quandle cocycle invariant, we give a method to detect the necessity of Reidemeister moves of type II between two given virtual-knot (or classical-knot) diagrams. As an application, we show that for any virtual-knot diagram [Formula: see text], there exists a diagram [Formula: see text] representing the same virtual-knot such that any sequence of generalized Reidemeister moves between them includes at least one Reidemeister move of type II.

scholarly journals A relation between the crossing number and the height of a knotoid

2021 ◽  
pp. 2150040
Author(s):  
Philipp Korablev ◽  
Vladimir Tarkaev
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Crossing Number ◽  
Reidemeister Moves ◽  
Bracket Polynomial ◽  
Knot Diagrams ◽  
Minimal Diagram

Knotoids are open ended knot diagrams regarded up to Reidemeister moves and isotopies. The notion is introduced by Turaev in 2012. Two most important numeric characteristics of a knotoid are the crossing number and the height. The latter is the least number of intersections between a diagram and an arc connecting its endpoints, where the minimum is taken over all representative diagrams and all such arcs which are disjoint from crossings. In the paper, we answer the question: are there any relations between the crossing number and the height of a knotoid. We prove that the crossing number of a knotoid is greater than or equal to twice the height of the knotoid. Combining the inequality with known lower bounds of the height we obtain a lower bounds of the crossing number of a knotoid via the extended bracket polynomial, the affine index polynomial and the arrow polynomial of the knotoid. As an application of our result we prove an upper bound for the length of a bridge in a minimal diagram of a classical knot: the number of crossings in a minimal diagram of a knot is greater than or equal to three times the length of a longest bridge in the diagram.

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.

REIDEMEISTER MOVES FOR SURFACE ISOTOPIES AND THEIR INTERPRETATION AS MOVES TO MOVIES

1993 ◽  
Vol 02 (03) ◽  
pp. 251-284 ◽  
Author(s):  
J. SCOTT CARTER ◽  
MASAHICO SAITO
Keyword(s):  
Cross Sections ◽  
Height Function ◽  
The Other ◽  
Projection Direction ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Fixed Direction ◽  
The Cross ◽  
Knot Diagrams

A movie description of a surface embedded in 4-space is a sequence of knot and link diagrams obtained from a projection of the surface to 3-space by taking 2-dimensional cross sections perpendicular to a fixed direction. In the cross sections, an immersed collection of curves appears, and these are lifted to knot diagrams by using the projection direction from 4-space. We give a set of 15 moves to movies (called movie moves) such that two movies represent isotopic surfaces if and only if there is a sequence of moves from this set that takes one to the other. This result generalizes the Roseman moves which are moves on projections where a height function has not been specified. The first 7 of the movie moves are height function parametrized versions of those given by Roseman. The remaining 8 are moves in which the topology of the projection remains unchanged.

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.

LOWER BOUNDS FOR THE NORMALIZED HEIGHT AND NON-DENSE SUBSETS OF SUBVARIETIES OF ABELIAN VARIETIES

2010 ◽  
Vol 06 (03) ◽  
pp. 471-499 ◽  
Author(s):  
EVELINA VIADA
Keyword(s):  
Lower Bound ◽  
Elliptic Curve ◽  
Abelian Variety ◽  
Lower Bounds ◽  
Finite Rank ◽  
Height Function ◽  
Algebraic Numbers ◽  
Algebraic Subgroup ◽  
Rank Zero ◽  
The Third

This work is the third part of a series of papers. In the first two, we considered curves and varieties in a power of an elliptic curve. Here, we deal with subvarieties of an abelian variety in general. Let V be a proper irreducible subvariety of dimension d in an abelian variety A, both defined over the algebraic numbers. We say that V is weak-transverse if V is not contained in any proper algebraic subgroup of A, and transverse if it is not contained in any translate of such a subgroup. Assume a conjectural lower bound for the normalized height of V. Then, for V transverse, we prove that the algebraic points of bounded height of V which lie in the union of all algebraic subgroups of A of codimension at least d + 1 translated by the points close to a subgroup Γ of finite rank, are non-Zariski-dense in V. If Γ has rank zero, it is sufficient to assume that V is weak-transverse. The notion of closeness is defined using a height function.

scholarly journals (1, 2) AND WEAK (1, 3) HOMOTOPIES ON KNOT PROJECTIONS

2013 ◽  
Vol 22 (14) ◽  
pp. 1350085 ◽  
Author(s):  
NOBORU ITO ◽  
YUSUKE TAKIMURA
Keyword(s):  
Equivalence Relation ◽  
Finite Sequence ◽  
Reidemeister Move ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Homotopy Classes ◽  
The Third ◽  
Reidemeister Moves ◽  
Necessary And Sufficient

In this paper, we obtain the necessary and sufficient condition that two knot projections are related by a finite sequence of the first and second flat Reidemeister moves (Theorem 2.2). We also consider an equivalence relation that is called weak (1, 3) homotopy. This equivalence relation occurs by the first flat Reidemeister move and one of the third flat Reidemeister moves. We introduce a map sending weak (1, 3) homotopy classes to knot isotopy classes (Sec. 3). Using the map, we determine which knot projections are trivialized under weak (1, 3) homotopy (Corollary 4.1).

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