Arc presentations of Montesinos links

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.

Paediatric flat foot and foot dimension in Central Anatolia

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.

Minimal grid diagrams of 11 crossing prime alternating knots

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.

Bisected vertex leveling of plane graphs: Braid index, arc index and delta diagrams

In this paper, we introduce a bisected vertex leveling of a plane graph. Using this planar embedding, we present elementary proofs of the well-known upper bounds in terms of the minimal crossing number on braid index [Formula: see text] and arc index [Formula: see text] for any knot or non-split link [Formula: see text], which are [Formula: see text] and [Formula: see text]. We also find a quadratic upper bound of the minimal crossing number of delta diagrams of [Formula: see text].

Stick number of spatial graphs

For a nontrivial knot [Formula: see text], Negami found an upper bound on the stick number [Formula: see text] in terms of its crossing number [Formula: see text] which is [Formula: see text]. Later, Huh and Oh utilized the arc index [Formula: see text] to present a more precise upper bound [Formula: see text]. Furthermore, Kim, No and Oh found an upper bound on the equilateral stick number [Formula: see text] as follows; [Formula: see text]. As a sequel to this research program, we similarly define the stick number [Formula: see text] and the equilateral stick number [Formula: see text] of a spatial graph [Formula: see text], and present their upper bounds as follows; [Formula: see text] [Formula: see text] where [Formula: see text] and [Formula: see text] are the number of edges and vertices of [Formula: see text], respectively, [Formula: see text] is the number of bouquet cut-components, and [Formula: see text] is the number of non-splittable components.

Mutation invariance of the arc index for some montesinos knots

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.

On the arc index of Kanenobu knots

In this paper, we calculate the Kauffman polynomials [Formula: see text] of Kanenobu knots [Formula: see text] with [Formula: see text] half twists and determine their spans on the variable [Formula: see text] completely. As an application, we determine the arc index of infinitely many Kanenobu knots. In particular, we give sharper lower bounds of the arc index of [Formula: see text] by using canonical cabling algorithm and the 2-cable [Formula: see text]-polynomials. Moreover, we give sharper upper bounds of the arc index of some Kanenobu knots by using their braid presentations.

On the arc index of cable links and Whitehead doubles

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.