Mosaic number of knots

2014 ◽  
Vol 23 (13) ◽  
pp. 1450069 ◽  
Author(s):  
Hwa Jeong Lee ◽  
Kyungpyo Hong ◽  
Ho Lee ◽  
Seungsang Oh
Keyword(s):  
Quantum System ◽  
Upper Bound ◽  
Crossing Number ◽  
System A ◽  
Definition Of ◽  
Hopf Link

Lomonaco and Kauffman developed knot mosaics to give a definition of a quantum knot system. This definition is intended to represent an actual physical quantum system. A knot n-mosaic is an n × n matrix of 11 kinds of specific mosaic tiles representing a knot or a link. The mosaic number m(K) of a knot K is the smallest integer n for which K is representable as a knot n-mosaic. In this paper, we establish an upper bound on the mosaic number of a knot or a link K in terms of the crossing number c(K). Let K be a nontrivial knot or a non-split link except the Hopf link. Then m(K) ≤ c(K) + 1. Moreover if K is prime and non-alternating except [Formula: see text] link, then m(K) ≤ c(K) - 1.

scholarly journals Upper bound on the total number of knot n-mosaics

2014 ◽  
Vol 23 (13) ◽  
pp. 1450065 ◽  
Author(s):  
Kyungpyo Hong ◽  
Seungsang Oh ◽  
Ho Lee ◽  
Hwa Jeong Lee
Keyword(s):  
Quantum System ◽  
Upper Bound ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Exact Number ◽  
System A ◽  
Definition Of

Lomonaco and Kauffman introduced a knot mosaic system to give a definition of a quantum knot system which can be viewed as a blueprint for the construction of an actual physical quantum system. A knot n-mosaic is an n × n matrix of 11 kinds of specific mosaic tiles representing a knot or a link by adjoining properly that is called suitably connected. Dn denotes the total number of all knot n-mosaics. Already known is that D1 = 1, D2 = 2 and D3 = 22. In this paper we establish the lower and upper bounds on Dn[Formula: see text] and find the exact number of D4 = 2594.

scholarly journals Triple-crossing number and moves on triple-crossing link diagrams

2019 ◽  
Vol 28 (11) ◽  
pp. 1940001 ◽  
Author(s):  
Colin Adams ◽  
Jim Hoste ◽  
Martin Palmer
Keyword(s):  
Triple Point ◽  
Crossing Number ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Triple Points ◽  
Definition Of ◽  
Hopf Link

Every link in the 3-sphere has a projection to the plane where the only singularities are pairwise transverse triple points. The associated diagram, with height information at each triple point, is a triple-crossing diagram of the link. We give a set of diagrammatic moves on triple-crossing diagrams analogous to the Reidemeister moves on ordinary diagrams. The existence of [Formula: see text]-crossing diagrams for every [Formula: see text] greater than one allows the definition of the [Formula: see text]-crossing number. We prove that for any nontrivial, nonsplit link, other than the Hopf link, its triple-crossing number is strictly greater than its quintuple-crossing number.

scholarly journals Period and toroidal knot mosaics

2017 ◽  
Vol 26 (05) ◽  
pp. 1750031 ◽  
Author(s):  
Seungsang Oh ◽  
Kyungpyo Hong ◽  
Ho Lee ◽  
Hwa Jeong Lee ◽  
Mi Jeong Yeon
Keyword(s):  
Quantum System ◽  
Prime Number ◽  
Growth Rates ◽  
Exact Enumeration ◽  
Common Features ◽  
System A ◽  
Number Formula ◽  
And Mathematics ◽  
Definition Of ◽  
Positive Integers

Knot mosaic theory was introduced by Lomonaco and Kauffman in the paper on ‘Quantum knots and mosaics’ to give a precise and workable definition of quantum knots, intended to represent an actual physical quantum system. A knot [Formula: see text]-mosaic is an [Formula: see text] matrix whose entries are eleven mosaic tiles, representing a knot or a link by adjoining properly. In this paper, we introduce two variants of knot mosaics: period knot mosaics and toroidal knot mosaics, which are common features in physics and mathematics. We present an algorithm producing the exact enumeration of period knot [Formula: see text]-mosaics for any positive integers [Formula: see text] and [Formula: see text], toroidal knot [Formula: see text]-mosaics for co-prime integers [Formula: see text] and [Formula: see text], and furthermore toroidal knot [Formula: see text]-mosaics for a prime number [Formula: see text]. We also analyze the asymptotics of the growth rates of their cardinality.

scholarly journals Zur exakten Behandlung von kollektiven Rotationen Teil A: Theorie / On the Exact Treatment of Collective Rotations Part A: Theory

1973 ◽  
Vol 28 (2) ◽  
pp. 206-215
Author(s):  
Hanns Ruder
Keyword(s):  
Coordinate System ◽  
Perturbation Expansion ◽  
Rotational Energy ◽  
The Body ◽  
System A ◽  
N Body Problem ◽  
Definition Of ◽  
Fixed System ◽  
Body Fixed Coordinate System ◽  
Groundstate Energy

Basic in the treatment of collective rotations is the definition of a body-fixed coordinate system. A kinematical method is derived to obtain the Hamiltonian of a n-body problem for a given definition of the body-fixed system. From this exact Hamiltonian, a consequent perturbation expansion in terms of the total angular momentum leads to two exact expressions: one for the collective rotational energy which has to be added to the groundstate energy in this order of perturbation and a second one for the effective inertia tensor in the groundstate. The discussion of these results leads to two criteria how to define the best body-fixed coordinate system, namely a differential equation and a variational principle. The equivalence of both is shown.

XV.—The Placoderm Fish Rhachiosteus pterygiatus Gross and its Relationships

1966 ◽  
Vol 66 (15) ◽  
pp. 377-392 ◽  
Author(s):  
Roger S. Miles
Keyword(s):  
Sensory System ◽  
System A ◽  
The Family ◽  
Definition Of

SynopsisThe holotype and only known specimen of Rhachiosteus pterygiatus Gross is partially redescribed and new restorations are given. Attention is drawn to important points in its osteology and the possible development of a cutaneous sensory system. A definition of the family Rhachiosteidsæ Stensiö is given. This family differs from all other described groups of euarthrodires in the lack of posterior lateral and posterior dorsolateral flank plates. Rhachiosteus is a pachyosteomorph brachythoracid, as defined in the text, and may be fairly closely related in some way to the (coccosteomorph) family Coccosteidsæ. There is no indication that it is closely related to any other known pachyosteomorph, or to other groups of arthrodires, such as the Rhenanida and Ptyctodontida, in which there are no posterior flank plates.

Stick numbers of Montesinos knots

2021 ◽  
pp. 2150013
Author(s):  
Hwa Jeong Lee ◽  
Sungjong No ◽  
Seungsang Oh
Keyword(s):  
Upper Bound ◽  
Crossing Number ◽  
Number Formula ◽  
Knots And Links

Negami found an upper bound on the stick number [Formula: see text] of a nontrivial knot [Formula: see text] in terms of the minimal crossing number [Formula: see text]: [Formula: see text]. Huh and Oh found an improved upper bound: [Formula: see text]. Huh, No and Oh proved that [Formula: see text] for a [Formula: see text]-bridge knot or link [Formula: see text] with at least six crossings. As a sequel to this study, we present an upper bound on the stick number of Montesinos knots and links. Let [Formula: see text] be a knot or link which admits a reduced Montesinos diagram with [Formula: see text] crossings. If each rational tangle in the diagram has five or more index of the related Conway notation, then [Formula: see text]. Furthermore, if [Formula: see text] is alternating, then we can additionally reduce the upper bound by [Formula: see text].

scholarly journals Bank Stability Measures in Dual Banking System: A Critical Review

2019 ◽  
Vol 5 (2) ◽  
pp. 59
Author(s):  
Norzitah Abdul Karim ◽  
Amirul Afiff Muhamat ◽  
Azreen Roslan ◽  
Sharifah Faigah Syed Alwi ◽  
Mohamad Nizam Jaafar
Keyword(s):  
Global Financial Crisis ◽  
Financial Institution ◽  
Banking System ◽  
Banking Supervision ◽  
Basel Committee ◽  
System A ◽  
Overall Stability ◽  
Regulatory Bodies ◽  
Banking Stability ◽  
Definition Of

The 2007-2009 Global Financial Crisis showed that despite reported as ‘healthy’ financial institution prior to crisis had indeed suffered many problems including liquidity during the crisis. Thus, there is confusion on the healthy financial institutions, leading to loss of confidence on the overall stability of the banking system. Thus, there is an urgent need to review the current measures of financial as well as banking stability. This paper aims to look at the definition of ‘stability’ used in the academic researches and by different regulatory bodies, like International Monetary Fund, Basel Committee for Banking Supervision (BCBS) and central banks in selected countries with dual banking systems. It is then, critically review indicators used as measures of financial as well as banking stability. This review is hope to identify areas of strengths as well as weaknesses of the current measures of stability and serves as foundation for further research in future.

CONSTRUCTING ALGEBRAIC LINKS FOR LOW EDGE NUMBERS

2005 ◽  
Vol 14 (06) ◽  
pp. 713-733 ◽  
Author(s):  
CYNTHIA L. McCABE
Keyword(s):  
Upper Bound ◽  
Crossing Number

A method is given for economically constructing any algebraic knot or link K. This construction, which involves tree diagrams, gives a new upper bound for the edge number of K that is proven to be at most twice the crossing number of K. Furthermore, it realizes a minimal-crossing projection.

An Improved Upper Bound on the Crossing Number of the Hypercube

2003 ◽  
pp. 230-236 ◽  
Author(s):  
Luerbio Faria ◽  
Celina M. Herrera de Figueiredo ◽  
Ondrej Sýkora ◽  
Imrich Vrt’o
Keyword(s):  
Upper Bound ◽  
Crossing Number
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