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.

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].

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

An improved upper bound on the crossing number of the hypercube

2008 ◽  
Vol 59 (2) ◽  
pp. 145-161 ◽  
Author(s):  
Luerbio Faria ◽  
Celina Miraglia Herrera de Figueiredo ◽  
Ondrej Sýkora ◽  
Imrich Vrt'o
Keyword(s):  
Upper Bound ◽  
Crossing Number

ALTERNATING HAMILTON CYCLES WITH MINIMUM NUMBER OF CROSSINGS IN THE PLANE

2000 ◽  
Vol 10 (01) ◽  
pp. 73-78 ◽  
Author(s):  
ATSUSHI KANEKO ◽  
M. KANO ◽  
KIYOSHI YOSHIMOTO
Keyword(s):  
Upper Bound ◽  
Time Algorithm ◽  
Hamilton Cycle ◽  
Crossing Number ◽  
Hamilton Cycles ◽  
Line Segments ◽  
Straight Line ◽  
Minimum Number ◽  
Disjoint Sets

Let X and Y be two disjoint sets of points in the plane such that |X|=|Y| and no three points of X ∪ Y are on the same line. Then we can draw an alternating Hamilton cycle on X∪Y in the plane which passes through alternately points of X and those of Y, whose edges are straight-line segments, and which contains at most |X|-1 crossings. Our proof gives an O(n2 log n) time algorithm for finding such an alternating Hamilton cycle, where n =|X|. Moreover we show that the above upper bound |X|-1 on crossing number is best possible for some configurations.

scholarly journals Crossings in Grid Drawings

10.37236/3025 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Vida Dujmović ◽  
Pat Morin ◽  
Adam Sheffer
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Geometric Graph ◽  
Upper Bounds ◽  
Crossing Number ◽  
Geometric Graphs ◽  
Vertex Sets

We prove tight crossing number inequalities for geometric graphs whose vertex sets are taken from a $d$-dimensional grid of volume $N$ and give applications of these inequalities to counting the number of crossing-free geometric graphs that can be drawn on such grids.In particular, we show that any geometric graph with $m\geq 8N$ edges and with vertices on a 3D integer grid of volume $N$, has $\Omega((m^2/N)\log(m/N))$ crossings. In $d$-dimensions, with $d\ge 4$, this bound becomes $\Omega(m^2/N)$. We provide matching upper bounds for all $d$. Finally, for $d\ge 4$ the upper bound implies that the maximum number of crossing-free geometric graphs with vertices on some $d$-dimensional grid of volume $N$ is $N^{\Theta(N)}$. In 3 dimensions it remains open to improve the trivial bounds, namely, the $2^{\Omega(N)}$ lower bound and the $N^{O(N)}$ upper bound.

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 On Bipartite Distinct Distances in the Plane

10.37236/9687 ◽  
2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Surya Mathialagan
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Crossing Number ◽  
Minimum Number

Given sets $\mathcal{P}, \mathcal{Q} \subseteq \mathbb{R}^2$ of sizes $m$ and $n$ respectively, we are interested in the number of distinct distances spanned by $\mathcal{P} \times \mathcal{Q}$. Let $D(m, n)$ denote the minimum number of distances determined by sets in $\mathbb{R}^2$ of sizes $m$ and $n$ respectively, where $m \leq n$. Elekes showed that $D(m, n) = O(\sqrt{mn})$ when $m \leqslant n^{1/3}$. For $m \geqslant n^{1/3}$, we have the upper bound $D(m, n) = O(n/\sqrt{\log n})$ as in the classical distinct distances problem.In this work, we show that Elekes' construction is tight by deriving the lower bound of $D(m, n) = \Omega(\sqrt{mn})$ when $m \leqslant n^{1/3}$. This is done by adapting Székely's crossing number argument. We also extend the Guth and Katz analysis for the classical distinct distances problem to show a lower bound of $D(m, n) = \Omega(\sqrt{mn}/\log n)$ when $m \geqslant n^{1/3}$.

scholarly journals ON THE CROSSING NUMBER OF THE CARTESIAN PRODUCT OF A SUNLET GRAPH AND A STAR GRAPH

2019 ◽  
Vol 100 (1) ◽  
pp. 5-12
Author(s):  
MICHAEL HAYTHORPE ◽  
ALEX NEWCOMBE
Keyword(s):  
Upper Bound ◽  
Cartesian Product ◽  
Crossing Number ◽  
Star Graph ◽  
Cartesian Products ◽  
Crossing Numbers ◽  
First Time

The exact crossing number is only known for a small number of families of graphs. Many of the families for which crossing numbers have been determined correspond to cartesian products of two graphs. Here, the cartesian product of the sunlet graph, denoted ${\mathcal{S}}_{n}$, and the star graph, denoted $K_{1,m}$, is considered for the first time. It is proved that the crossing number of ${\mathcal{S}}_{n}\Box K_{1,2}$ is $n$, and the crossing number of ${\mathcal{S}}_{n}\Box K_{1,3}$ is $3n$. An upper bound for the crossing number of ${\mathcal{S}}_{n}\Box K_{1,m}$ is also given.

scholarly journals CROSSING NUMBER OF LINKS FORMED BY EDGES OF A TRIANGULATION

2003 ◽  
Vol 12 (03) ◽  
pp. 281-286 ◽  
Author(s):  
SIMON A. KING
Keyword(s):  
Exponential Function ◽  
Upper Bound ◽  
Crossing Number ◽  
Numerical Invariant

We study the crossing number of links that are formed by edges of a triangulation [Formula: see text] of S3 with n tetrahedra. We show that the crossing number is bounded from above by an exponential function of n2. In general, this bound can not be replaced by a subexponential bound. However, if [Formula: see text] is polytopal (resp. shellable) then there is a quadratic (resp. biquadratic) upper bound in n for the crossing number. In our proof, we use a numerical invariant [Formula: see text], called polytopality, introduced by the author.

ON FINITENESS OF VASSILIEV INVARIANTS AND A PROOF OF THE LIN-WANG CONJECTURE VIA BRAIDING POLYNOMIALS

2001 ◽  
Vol 10 (05) ◽  
pp. 769-780 ◽  
Author(s):  
A. Stoimenow
Keyword(s):  
Upper Bound ◽  
Special Class ◽  
Crossing Number ◽  
New Approach ◽  
Vassiliev Invariants ◽  
A Value ◽  
Vassiliev Invariant

Using the new approach of braiding sequences we give a proof of the Lin-Wing conjecture, stating that a Vassiliev invariant ν of degree k has a value Oν (c(K)k) on a knot K, where c(K) is the crossing number of K and Oν depends on ν only. We extend our method to give a quadratic upper bound in k for the crossing number of alternating/positive knots, the values on which suffice to determine uniquely a Vassiliev invariant of degree k. This also makes orientation and mutation sensitivity of Vassiliev invariants decidable by testing them on alternating/positive knots/mutants only. We give an exponential upper bound for the number of Vassiliev invariants on a special class of closed braids.

Sign in / Sign up

Export Citation Format

Share Document