scholarly journals Lattice stick number of spatial graphs

2018 ◽  
Vol 27 (08) ◽  
pp. 1850048
Author(s):  
Hyungkee Yoo ◽  
Chaeryn Lee ◽  
Seungsang Oh
Keyword(s):  
Upper Bound ◽  
Crossing Number ◽  
Cubic Lattice ◽  
Spatial Graphs ◽  
Number Formula

The lattice stick number of knots is defined to be the minimal number of straight sticks in the cubic lattice required to construct a lattice stick presentation of the knot. We similarly define the lattice stick number [Formula: see text] of spatial graphs [Formula: see text] with vertices of degree at most six (necessary for embedding into the cubic lattice), and present an upper bound in terms of the crossing number [Formula: see text] [Formula: see text] where [Formula: see text] has [Formula: see text] edges, [Formula: see text] vertices, [Formula: see text] cut-components, [Formula: see text] bouquet cut-components, and [Formula: see text] knot components.

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.

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].

Circuit presentation and lattice stick number with exactly four z-sticks

2018 ◽  
Vol 27 (08) ◽  
pp. 1850046
Author(s):  
Hyoungjun Kim ◽  
Sungjong No
Keyword(s):  
Upper Bound ◽  
Line Segment ◽  
Disjoint Union ◽  
Lattice Plane ◽  
Cambridge Philos ◽  
Line Segments ◽  
Cubic Lattice ◽  
Straight Line ◽  
Two Component ◽  
Number Formula

The lattice stick number [Formula: see text] of a link [Formula: see text] is defined to be the minimal number of straight line segments required to construct a stick presentation of [Formula: see text] in the cubic lattice. Hong, No and Oh [Upper bound on lattice stick number of knots, Math. Proc. Cambridge Philos. Soc. 155 (2013) 173–179] found a general upper bound [Formula: see text]. A rational link can be represented by a lattice presentation with exactly 4 [Formula: see text]-sticks. An [Formula: see text]-circuit is the disjoint union of [Formula: see text] arcs in the lattice plane [Formula: see text]. An [Formula: see text]-circuit presentation is an embedding obtained from the [Formula: see text]-circuit by connecting each [Formula: see text] pair of vertices with one line segment above the circuit. By using a two-circuit presentation, we can easily find the lattice presentation with exactly four [Formula: see text]-sticks. In this paper, we show that an upper bound for the lattice stick number of rational [Formula: see text]-links realized with exactly four [Formula: see text]-sticks is [Formula: see text]. Furthermore, it is [Formula: see text] if [Formula: see text] is a two-component link.

scholarly journals The ropelengths of knots are almost linear in terms of their crossing numbers

2019 ◽  
Vol 28 (14) ◽  
pp. 1950085
Author(s):  
Yuanan Diao ◽  
Claus Ernst ◽  
Attila Por ◽  
Uta Ziegler
Keyword(s):  
Upper Bound ◽  
Plane Graph ◽  
Crossing Number ◽  
Divide And Conquer ◽  
Plane Graphs ◽  
Cubic Lattice ◽  
Crossing Numbers ◽  
Total Length

For a knot or link [Formula: see text], let [Formula: see text] be the ropelength of [Formula: see text] and [Formula: see text] be the crossing number of [Formula: see text]. In this paper, we show that there exists a constant [Formula: see text] such that [Formula: see text] for any [Formula: see text], i.e. the upper bound of the ropelength of any knot is almost linear in terms of its minimum crossing number. This result is a significant improvement over the best known upper bound established previously, which is of the form [Formula: see text]. The proof is based on a divide-and-conquer approach on 4-regular plane graphs: a 4-regular plane graph of [Formula: see text] is first repeatedly subdivided into many small subgraphs and then reconstructed from these small subgraphs on the cubic lattice with its topology preserved with a total length of the order [Formula: see text]. The result then follows since a knot can be recovered from a graph that is topologically equivalent to a regular projection of it (which is a 4-regular plane graph).

scholarly journals Upper bound on lattice stick number of knots

2013 ◽  
Vol 155 (1) ◽  
pp. 173-179 ◽  
Author(s):  
KYUNGPYO HONG ◽  
SUNGJONG NO ◽  
SEUNGSANG OH
Keyword(s):  
Upper Bound ◽  
Crossing Number ◽  
Line Segments ◽  
Cubic Lattice ◽  
Straight Line ◽  
Trefoil Knot

AbstractThe lattice stick number sL(K) of a knot K is defined to be the minimal number of straight line segments required to construct a stick presentation of K in the cubic lattice. In this paper, we find an upper bound on the lattice stick number of a nontrivial knot K, except the trefoil knot, in terms of the minimal crossing number c(K) which is sL(K) ≤ 3c(K) + 2. Moreover if K is a non-alternating prime knot, then sL(K) ≤ 3c(K) − 4.

Crossing number bounds in knot mosaics

2018 ◽  
Vol 27 (10) ◽  
pp. 1850056 ◽  
Author(s):  
Hugh Howards ◽  
Andrew Kobin
Keyword(s):  
Upper Bound ◽  
Crossing Number ◽  
Quantum States ◽  
Number Formula

Knot mosaics are used to model physical quantum states. The mosaic number of a knot is the smallest integer [Formula: see text] such that the knot can be represented as a knot [Formula: see text]-mosaic. In this paper, we establish an upper bound for the crossing number of a knot in terms of the mosaic number. Given an [Formula: see text]-mosaic and any knot [Formula: see text] that is represented on the mosaic, its crossing number [Formula: see text] is bounded above by [Formula: see text] if [Formula: see text] is odd, and by [Formula: see text] if [Formula: see text] is even. In the process, we develop a useful new tool called the mosaic complement.

Incoherent nullification of torus knots and links

2019 ◽  
Vol 28 (05) ◽  
pp. 1950033
Author(s):  
Zac Bettersworth ◽  
Claus Ernst
Keyword(s):  
Upper Bound ◽  
Torus Knots ◽  
Number Formula ◽  
Knots And Links

In the paper, we study the incoherent nullification number [Formula: see text] of knots and links. We establish an upper bound on the incoherent nullification number of torus knots and links and conjecture that this upper bound is the actual incoherent nullification number of this family. Finally, we establish the actual incoherent nullification number of particular subfamilies of torus knots and links.

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

Optimal rendezvous on a line by location-aware robots in the presence of spies

2021 ◽  
Author(s):  
Huda Chuangpishit ◽  
Jurek Czyzowicz ◽  
Ryan Killick ◽  
Evangelos Kranakis ◽  
Danny Krizanc
Keyword(s):  
Mobile Robots ◽  
Upper Bound ◽  
Competitive Ratio ◽  
Central Authority ◽  
Location Aware ◽  
Number Formula ◽  
Infinite Line ◽  
Optimal Rendezvous ◽  
Gps Devices ◽  
Time Required

A set of mobile robots is placed at arbitrary points of an infinite line. The robots are equipped with GPS devices and they may communicate their positions on the line to a central authority. The collection contains an unknown subset of “spies”, i.e., byzantine robots, which are indistinguishable from the non-faulty ones. The set of the non-faulty robots needs to rendezvous in the shortest possible time in order to perform some task, while the byzantine robots may try to delay their rendezvous for as long as possible. The problem facing a central authority is to determine trajectories for all robots so as to minimize the time until all the non-faulty robots have met. The trajectories must be determined without knowledge of which robots are faulty. Our goal is to minimize the competitive ratio between the time required to achieve the first rendezvous of the non-faulty robots and the time required for such a rendezvous to occur under the assumption that the faulty robots are known at the start. In this paper, we give rendezvous algorithms with bounded competitive ratio, where the central authority is informed only of the set of initial robot positions, without knowing which ones or how many of them are faulty. In general, regardless of the number of faults [Formula: see text] it can be shown that there is an algorithm with bounded competitive ratio. Further, we are able to give a rendezvous algorithm with optimal competitive ratio provided that the number [Formula: see text] of faults is strictly less than [Formula: see text]. Note, however, that in general this algorithm does not give an estimate on the actual value of the competitive ratio. However, when an upper bound on the number of byzantine robots is known to the central authority, we can provide algorithms with constant competitive ratios and in some instances we are able to show that these algorithms are optimal. Moreover, in the cases where the number of faults is either [Formula: see text] or [Formula: see text] we are able to compute the competitive ratio of an optimal rendezvous algorithm, for a small number of robots.

Sign in / Sign up

Export Citation Format

Share Document