scholarly journals Crosscap numbers of alternating knots via unknotting splices

2020 ◽  
Vol 31 (07) ◽  
pp. 2050057 ◽  
Author(s):  
Thomas Kindred
Keyword(s):  
Upper Bound ◽  
Number Formula ◽  
Knot Diagrams

Ito–Takimura recently defined a splice-unknotting number [Formula: see text] for knot diagrams. They proved that this number provides an upper bound for the crosscap number of any prime knot, asking whether equality holds in the alternating case. We answer their question in the affirmative. (Ito has independently proven the same result.) As an application, we compute the crosscap numbers of all prime alternating knots through at least 13 crossings, using Gauss codes.

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].

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.

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.

scholarly journals An Upper Bound on the k-Modem Illumination Problem

2015 ◽  
Vol 25 (04) ◽  
pp. 299-308
Author(s):  
Frank Duque ◽  
Carlos Hidalgo-Toscano
Keyword(s):  
Upper Bound ◽  
Line Segment ◽  
Time Algorithm ◽  
Fixed Number ◽  
Light Sources ◽  
Wireless Devices ◽  
Orthogonal Polygon ◽  
Illumination Problem ◽  
Number Formula ◽  
Interior Points

A variation on the classical polygon illumination problem was introduced in [Aichholzer et al. EuroCG’09]. In this variant light sources are replaced by wireless devices called [Formula: see text]-modems, which can penetrate a fixed number [Formula: see text], of “walls”. A point in the interior of a polygon is “illuminated” by a [Formula: see text]-modem if the line segment joining them intersects at most [Formula: see text] edges of the polygon. It is easy to construct polygons of [Formula: see text] vertices where the number of [Formula: see text]-modems required to illuminate all interior points is [Formula: see text]. However, no non-trivial upper bound is known. In this paper we prove that the number of kmodems required to illuminate any polygon of [Formula: see text] vertices is [Formula: see text]. For the cases of illuminating an orthogonal polygon or a set of disjoint orthogonal segments, we give a tighter bound of [Formula: see text]. Moreover, we present an [Formula: see text] time algorithm to achieve this bound.

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.

Geodesic graphs for a homotopy class of virtual knots

2017 ◽  
Vol 26 (13) ◽  
pp. 1750090
Author(s):  
Sumiko Horiuchi ◽  
Yoshiyuki Ohyama
Keyword(s):  
Homotopy Class ◽  
Finite Sequence ◽  
Dimensional Lattice ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Minimum Number ◽  
Vertex Set ◽  
Number Formula ◽  
Lattice Graph ◽  
Knot Diagrams

We consider a local move, denoted by [Formula: see text], on knot diagrams or virtual knot diagrams.If two (virtual) knots [Formula: see text] and [Formula: see text] are transformed into each other by a finite sequence of [Formula: see text] moves, the [Formula: see text] distance between [Formula: see text] and [Formula: see text] is the minimum number of times of [Formula: see text] moves needed to transform [Formula: see text] into [Formula: see text]. By [Formula: see text], we denote the set of all (virtual) knots which can be transformed into a (virtual) knot [Formula: see text] by [Formula: see text] moves. A geodesic graph for [Formula: see text] is the graph which satisfies the following: The vertex set consists of (virtual) knots in [Formula: see text] and for any two vertices [Formula: see text] and [Formula: see text], the distance on the graph from [Formula: see text] to [Formula: see text] coincides with the [Formula: see text] distance between [Formula: see text] and [Formula: see text]. When we consider virtual knots and a crossing change as a local move [Formula: see text], we show that the [Formula: see text]-dimensional lattice graph for any given natural number [Formula: see text] and any tree are geodesic graphs for [Formula: see text].

scholarly journals Quadratic nonresidues below the Burgess bound

2017 ◽  
Vol 13 (03) ◽  
pp. 751-759 ◽  
Author(s):  
William D. Banks ◽  
Victor Z. Guo
Keyword(s):  
Prime Number ◽  
Upper Bound ◽  
Legendre Symbol ◽  
Natural Numbers ◽  
Number Formula

For any odd prime number [Formula: see text], let [Formula: see text] be the Legendre symbol, and let [Formula: see text] be the sequence of positive nonresidues modulo [Formula: see text], i.e. [Formula: see text] for each [Formula: see text]. In 1957, Burgess showed that the upper bound [Formula: see text] holds for any fixed [Formula: see text]. In this paper, we prove that the stronger bound [Formula: see text] holds for all odd primes [Formula: see text] provided that [Formula: see text] where the implied constants are absolute. For fixed [Formula: see text], we also show that there is a number [Formula: see text] such that for all odd primes [Formula: see text], there are [Formula: see text] natural numbers [Formula: see text] with [Formula: see text] provided that [Formula: see text]

The Critical Node Problem Based on Connectivity Index and Properties of Components on Trees

2020 ◽  
pp. 2050041
Author(s):  
Xiucui Guan ◽  
Chao Liu ◽  
Qiao Zhang
Keyword(s):  
Upper Bound ◽  
Connectivity Index ◽  
Component Size ◽  
Number Of Components ◽  
Critical Node ◽  
Residual Graph ◽  
Number Formula ◽  
The One ◽  
Critical Node Problem ◽  
Node Problem

We deal with the critical node problem (CNP) in a graph [Formula: see text], in which a given number [Formula: see text] of nodes are removed to minimize the connectivity of the residual graph in some sense. Several ways to minimize some connectivity measurement have been proposed, including minimizing the connectivity index(MinCI), maximizing the number of components, minimizing the maximal component size. We propose two classes of CNPs by combining the above measurements together. The objective is to minimize the sum of connectivity indexes and the total degrees in the residual graph. The CNP with an upper-bound [Formula: see text] on the maximal component size is denoted by MSCID-CS and the one with an extra upper-bound [Formula: see text] on the number of components is denoted by MSCID-CSN. They are generalizations of the MinCI, which has been shown NP-hard for general graphs. In particular, we study the case where [Formula: see text] is a tree. Two dynamic programming algorithms are proposed to solve the two classes of CNPs. The time complexities of the algorithms for MSCID-CS and MSCID-CSN are [Formula: see text] and [Formula: see text], respectively, where [Formula: see text] is the number of nodes in [Formula: see text]. Computational experiments are presented which show the effectiveness of the algorithms.

Upper bounds for the sum of Laplacian eigenvalues of a graph and Brouwer’s conjecture

2019 ◽  
Vol 11 (02) ◽  
pp. 1950028 ◽  
Author(s):  
Hilal A. Ganie ◽  
S. Pirzada ◽  
Rezwan Ul Shaban ◽  
X. Li
Keyword(s):  
Upper Bound ◽  
Clique Number ◽  
Upper Bounds ◽  
Simple Graph ◽  
Laplacian Eigenvalues ◽  
Number Formula ◽  
Split Graphs

Consider a simple graph [Formula: see text] of order [Formula: see text] and size [Formula: see text] having Laplacian eigenvalues [Formula: see text]. Let [Formula: see text] be the sum of [Formula: see text] largest Laplacian eigenvalues of [Formula: see text]. Brouwer conjectured that [Formula: see text] for all [Formula: see text]. We obtain an upper bound for [Formula: see text] in terms of the clique number [Formula: see text], the number of vertices [Formula: see text] and the non-negative integers [Formula: see text] associated to the structure of the graph [Formula: see text]. We show that the Brouwer’s conjecture holds true for some new families of graphs. We use the same technique to prove that the Brouwer’s conjecture is true for a subclass of split graphs (It is already known that Brouwer’s conjecture holds for split graphs).

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.

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