PRIME KNOTS WITH ARC INDEX UP TO 10

For the alternating knots or links, mutations do not change the arc index. In the case of nonalternating knots, some semi-alternating knots or links have this property. We mainly focus on the problem of mutation invariance of the arc index for nonalternating knots which are not semi-alternating. In this paper, we found families of infinitely many mutant pairs/triples of Montesinos knots with the same arc index.

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STICK AND RAMSEY NUMBERS OF TORUS LINKS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216513500272 ◽

2013 ◽

Vol 22 (07) ◽

pp. 1350027 ◽

Cited By ~ 1

Author(s):

MARIBETH JOHNSON ◽

STACY NICOLE MILLS ◽

ROLLAND TRAPP

Keyword(s):

General Formula ◽

Ramsey Number ◽

Ramsey Numbers ◽

Crossing Numbers ◽

Arc Index ◽

Bridge Number ◽

Torus Links

An algorithm that produces polygonal cable links is described, and applications discussed. In particular, the stick numbers of Tp,q torus links are shown to be 4p for 2p < q ≤ 3p, and it is shown that, in general, [Formula: see text]. Further, it is shown that the Ramsey number of a link is at least the sum of its arc index and bridge number. Using these results, we relate the Ramsey, stick and crossing numbers of torus links, showing [Formula: see text].

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On the arc index of cable links and Whitehead doubles

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216516500413 ◽

2016 ◽

Vol 25 (07) ◽

pp. 1650041 ◽

Cited By ~ 3

Author(s):

Hwa Jeong Lee ◽

Hideo Takioka

Keyword(s):

Upper Bounds ◽

Grid Diagrams ◽

Arc Index

In this paper, we construct an algorithm to produce canonical grid diagrams of cable links and Whitehead doubles, which give sharper upper bounds of the arc index of them. Moreover, we determine the arc index of [Formula: see text]-cable links and Whitehead doubles of all prime knots with up to eight crossings.

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Paediatric flat foot and foot dimension in Central Anatolia

BMC Pediatrics ◽

10.1186/s12887-021-02645-9 ◽

2021 ◽

Vol 21 (1) ◽

Author(s):

Serap Alsancak ◽

Senem Guner ◽

Enver Güven ◽

Ali Koray Özgün ◽

Yunis Akkaş ◽

...

Keyword(s):

Central Anatolia ◽

Foot Deformities ◽

Strongly Correlated ◽

Flat Foot ◽

Pes Cavus ◽

Pes Planus ◽

Foot Length ◽

Arc Index ◽

Arch Index ◽

The Relationship

Abstract Background Information on the foot structures of Central Anatolian children is limited. Foot structures of children aged 6–10 years were shown to be different according to sex and increasing age. Objective This study aimed to compare the foot anthropometric values by age and sex and collect the foot anthropometric data to reveal the relationship between pes planus and pes cavus in the arches of children according to age. Methods Footprints of 335 children (180 boys and 155 girls) aged 6–10 years were taken by the pedigraph method and evaluated using 18 different parameters. The TFL (Truncated foot length), FL (foot length), Arch Index, Chippaux Smirak Index, Staheli Arc Index, and foot rotation values of the children were examined. To examine the relationship between the parameters, normality values were examined. Independent samples t-test was used to analyze sex differences in terms of foot size and shape. Results Correlations between other parameters were determined using the correlations analysis method. TFL, metatarsal circumference, and FL were strongly correlated with age in the children. Foot rotation increased with body mass index in the girls compared to that in the boys. According to the evaluation results with the classification made with the Staheli arch index, 63.3% pes planus, 9.8% pes cavus and 27.7% of the normal arch structure were identified. Conclusions Planning shoe production accordingly will contribute to the development of healthy feet in children. This article focused on foot structures of in Central Anatolia and to identify early foot deformities in children. This study found that the length of the TFL was smaller in boys than in girls.

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Arc presentations of Montesinos links

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216521500474 ◽

2021 ◽

Vol 30 (07) ◽

Author(s):

Hwa Jeong Lee

Keyword(s):

Rational Numbers ◽

Crossing Number ◽

Arc Index

Let [Formula: see text] be a Montesinos link [Formula: see text] with positive rational numbers [Formula: see text] and [Formula: see text], each less than 1, and [Formula: see text] the minimal crossing number of [Formula: see text]. Herein, we construct arc presentations of [Formula: see text] with [Formula: see text], [Formula: see text] and [Formula: see text] arcs under some conditions for [Formula: see text], [Formula: see text] and [Formula: see text]. Furthermore, we determine the arc index of infinitely many Montesinos links.

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Minimal grid diagrams of 11 crossing prime alternating knots

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216520500765 ◽

2020 ◽

Vol 29 (11) ◽

pp. 2050076

Author(s):

Gyo Taek Jin ◽

Hwa Jeong Lee

Keyword(s):

Plane Curve ◽

Crossing Number ◽

Vertical Line ◽

Horizontal Line ◽

Line Segments ◽

Grid Diagrams ◽

Arc Index ◽

Closed Plane

The arc index of a knot is the minimal number of arcs in all arc presentations of the knot. An arc presentation of a knot can be shown in the form of a grid diagram which is a closed plane curve consisting of finitely many horizontal line segments and the same number of vertical line segments. The arc index of an alternating knot is its minimal crossing number plus two. In this paper, we give a list of minimal grid diagrams of the 11 crossing prime alternating knots obtained from arc presentations with 13 arcs.

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