On the arc index of cable links and Whitehead doubles

2016 ◽  
Vol 25 (07) ◽  
pp. 1650041 ◽  
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.

Minimal grid diagrams of 11 crossing prime alternating knots

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.

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

2018 ◽  
Vol 27 (08) ◽  
pp. 1850044
Author(s):  
Sungjong No ◽  
Seungsang Oh ◽  
Hyungkee Yoo
Keyword(s):  
Upper Bound ◽  
Plane Graph ◽  
Upper Bounds ◽  
Crossing Number ◽  
Index Formula ◽  
Plane Graphs ◽  
Planar Embedding ◽  
Braid Index ◽  
Arc Index

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].

scholarly journals Stick number of spatial graphs

2017 ◽  
Vol 26 (14) ◽  
pp. 1750100 ◽  
Author(s):  
Minjung Lee ◽  
Sungjong No ◽  
Seungsang Oh
Keyword(s):  
Research Program ◽  
Upper Bound ◽  
Upper Bounds ◽  
Crossing Number ◽  
Index Formula ◽  
Spatial Graph ◽  
Spatial Graphs ◽  
Number Formula ◽  
Arc Index

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.

scholarly journals PRIME KNOTS WHOSE ARC INDEX IS SMALLER THAN THE CROSSING NUMBER

2012 ◽  
Vol 21 (10) ◽  
pp. 1250103 ◽  
Author(s):  
GYO TAEK JIN ◽  
HWA JEONG LEE
Keyword(s):  
Crossing Number ◽  
Grid Diagrams ◽  
Arc Index

It is known that the arc index of alternating knots is the minimal crossing number plus two and the arc index of prime nonalternating knots is less than or equal to the minimal crossing number. We study some cases when the arc index is strictly less than the minimal crossing number. We also give minimal grid diagrams of some prime nonalternating knots with 13 crossings and 14 crossings whose arc index is the minimal crossing number minus one.

On the arc index of Kanenobu knots

2017 ◽  
Vol 26 (04) ◽  
pp. 1750015 ◽  
Author(s):  
Hwa Jeong Lee ◽  
Hideo Takioka
Keyword(s):  
Lower Bounds ◽  
Upper Bounds ◽  
Variable Formula ◽  
Arc Index

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.

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

Energy of spherical fuzzy graphs

2020 ◽  
pp. 321-332
Author(s):  
S. Yahya Mohamed ◽  
A. Mohamed Ali
Keyword(s):  
Adjacency Matrix ◽  
Upper Bounds ◽  
Fuzzy Graph ◽  
Lower And Upper Bounds ◽  
Fuzzy Graphs

In this paper, the notion of energy extended to spherical fuzzy graph. The adjacency matrix of a spherical fuzzy graph is defined and we compute the energy of a spherical fuzzy graph as the sum of absolute values of eigenvalues of the adjacency matrix of the spherical fuzzy graph. Also, the lower and upper bounds for the energy of spherical fuzzy graphs are obtained.

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