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.

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.

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.

Arc presentations of Montesinos links

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.

scholarly journals AN UPPER BOUND ON STICK NUMBER OF KNOTS

2011 ◽  
Vol 20 (05) ◽  
pp. 741-747 ◽  
Author(s):  
YOUNGSIK HUH ◽  
SEUNGSANG OH
Keyword(s):  
Upper Bound ◽  
Crossing Number ◽  
Arc Index

In 1991, Negami found an upper bound on the stick number s(K) of a nontrivial knot K in terms of crossing number c(K) which is s(K) ≤ 2c(K). In this paper we give a new upper bound in terms of arc index, and improve Negami's upper bound to [Formula: see text]. Moreover if K is a nonalternating prime knot, then [Formula: see text].

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.

A LIMITATION ON ALGORITHMS FOR CONSTRUCTING MINIMAL ARC-PRESENTATIONS FROM LINK DIAGRAMS

1998 ◽  
Vol 07 (04) ◽  
pp. 415-423 ◽  
Author(s):  
ELISABETTA BELTRAMI ◽  
PETER R. CROMWELL
Keyword(s):  
Crossing Number ◽  
Link Diagram ◽  
Link Diagrams ◽  
Arc Index

The arc index of a link is closely related to its crossing number. An algorithm is known which produces an arc-presentation by drawing the binding circle on a link diagram. When the algorithm can be applied and the link is alternating, the presentation is known to be minimal. Here we exhibit an example which shows that no such algorithm can be valid for all link diagrams.

scholarly journals PRIME KNOTS WITH ARC INDEX UP TO 11 AND AN UPPER BOUND OF ARC INDEX FOR NON-ALTERNATING KNOTS

2010 ◽  
Vol 19 (12) ◽  
pp. 1655-1672 ◽  
Author(s):  
GYO TAEK JIN ◽  
WANG KEUN PARK
Keyword(s):  
Half Plane ◽  
Upper Bound ◽  
Crossing Number ◽  
Boundary Line ◽  
Common Boundary ◽  
Arc Index

Every knot can be embedded in the union of finitely many half planes with a common boundary line in such a way that the portion of the knot in each half plane is a properly embedded arc. The minimal number of such half planes is called the arc index of the knot. We have identified all prime knots with arc index up to 11. We also proved that the crossing number is an upperbound of arc index for non-alternating knots. As a result the arc index is determined for prime knots up to twelve crossings.

On the Crossing Number of $K_{m,n}$

10.37236/1748 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Nagi H. Nahas
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number

The best lower bound known on the crossing number of the complete bipartite graph is : $$cr(K_{m,n}) \geq (1/5)(m)(m-1)\lfloor n/2 \rfloor \lfloor(n-1)/2\rfloor$$ In this paper we prove that: $$cr(K_{m,n}) \geq (1/5)m(m-1)\lfloor n/2 \rfloor \lfloor (n-1)/2 \rfloor + 9.9 \times 10^{-6} m^2n^2$$ for sufficiently large $m$ and $n$.

THE CYLINDRICAL CROSSING NUMBER OF ZERO DIVISOR GRAPHS

2020 ◽  
Vol 9 (8) ◽  
pp. 5901-5908
Author(s):  
M. Sagaya Nathan ◽  
J. Ravi Sankar
Keyword(s):  
Zero Divisor ◽  
Crossing Number
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