ON AN INEQUALITY BETWEEN UNKNOTTING NUMBER AND CROSSING NUMBER OF LINKS

2010 ◽  
Vol 19 (07) ◽  
pp. 893-903
Author(s):  
RYO HANAKI ◽  
JUNSUKE KANADOME
Keyword(s):  
Crossing Number ◽  
Minimal Diagram

It is well-known that for any link L, twice the unknotting number of L is less than or equal to the crossing number of L. Taniyama characterized the links which satisfy the equality. We characterize the links where twice the unknotting number is equal to the crossing number minus one. As a corollary, we show that for any link L with twice the unknotting number of L is greater than or equal to the crossing number of L minus two, every minimal diagram of L realizes the unknotting number.

scholarly journals A relation between the crossing number and the height of a knotoid

2021 ◽  
pp. 2150040
Author(s):  
Philipp Korablev ◽  
Vladimir Tarkaev
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Crossing Number ◽  
Reidemeister Moves ◽  
Bracket Polynomial ◽  
Knot Diagrams ◽  
Minimal Diagram

Knotoids are open ended knot diagrams regarded up to Reidemeister moves and isotopies. The notion is introduced by Turaev in 2012. Two most important numeric characteristics of a knotoid are the crossing number and the height. The latter is the least number of intersections between a diagram and an arc connecting its endpoints, where the minimum is taken over all representative diagrams and all such arcs which are disjoint from crossings. In the paper, we answer the question: are there any relations between the crossing number and the height of a knotoid. We prove that the crossing number of a knotoid is greater than or equal to twice the height of the knotoid. Combining the inequality with known lower bounds of the height we obtain a lower bounds of the crossing number of a knotoid via the extended bracket polynomial, the affine index polynomial and the arrow polynomial of the knotoid. As an application of our result we prove an upper bound for the length of a bridge in a minimal diagram of a classical knot: the number of crossings in a minimal diagram of a knot is greater than or equal to three times the length of a longest bridge in the diagram.

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

scholarly journals Join Products K2,3 + Cn

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 925
Author(s):  
Michal Staš
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number ◽  
Arithmetic Means ◽  
Minimum Number ◽  
Edge Crossings

The crossing number cr ( G ) of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main goal of the paper is to state the crossing number of the join product K 2 , 3 + C n for the complete bipartite graph K 2 , 3 , where C n is the cycle on n vertices. In the proofs, the idea of a minimum number of crossings between two distinct configurations in the various forms of arithmetic means will be extended. Finally, adding one more edge to the graph K 2 , 3 , we also offer the crossing number of the join product of one other graph with the cycle C n .

scholarly journals Improvement on the Crossing Number of Crossing-Critical Graphs

2020 ◽  
Author(s):  
János Barát ◽  
Géza Tóth
Keyword(s):  
Crossing Number ◽  
Critical Graphs ◽  
Minimum Number ◽  
Edge Crossings

AbstractThe crossing number of a graph G is the minimum number of edge crossings over all drawings of G in the plane. A graph G is k-crossing-critical if its crossing number is at least k, but if we remove any edge of G, its crossing number drops below k. There are examples of k-crossing-critical graphs that do not have drawings with exactly k crossings. Richter and Thomassen proved in 1993 that if G is k-crossing-critical, then its crossing number is at most $$2.5\, k+16$$ 2.5 k + 16 . We improve this bound to $$2k+8\sqrt{k}+47$$ 2 k + 8 k + 47 .

scholarly journals An Improvement of the Lower Bound on the Minimum Number of ≤k-Edges

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 525
Author(s):  
Javier Rodrigo ◽  
Susana Merchán ◽  
Danilo Magistrali ◽  
Mariló López
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
General Position ◽  
Crossing Number ◽  
Minimum Number

In this paper, we improve the lower bound on the minimum number of  ≤k-edges in sets of n points in general position in the plane when k is close to n2. As a consequence, we improve the current best lower bound of the rectilinear crossing number of the complete graph Kn for some values of n.

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].

Cyclic permutations and crossing numbers of join products of two symmetric graphs of order six

2019 ◽  
Vol 35 (2) ◽  
pp. 137-146
Author(s):  
STEFAN BEREZNY ◽  
MICHAL STAS ◽  
◽  
Keyword(s):  
Crossing Number ◽  
Crossing Numbers ◽  
Symmetric Graphs ◽  
Isolated Vertex ◽  
Join Of Graphs ◽  
Discrete Graph

The main purpose of this article is broaden known results concerning crossing numbers for join of graphs of order six. We give the crossing number of the join product G + Dn, where the graph G consists of one 5-cycle and of one isolated vertex, and Dn consists on n isolated vertices. The proof is done with the help of software that generates all cyclic permutations for a given number k, and creates a new graph COG for calculating the distances between all vertices of the graph. Finally, by adding some edges to the graph G, we are able to obtain the crossing numbers of the join product with the discrete graph Dn and with the path Pn on n vertices for other two graphs.

scholarly journals Lower Bounds on Virtual Crossing Number and Minimal Surface Genus

2011 ◽  
pp. 31-43 ◽  
Author(s):  
Kumud Bhandari ◽  
H. A. Dye ◽  
Louis H. Kauffman
Keyword(s):  
Minimal Surface ◽  
Lower Bounds ◽  
Crossing Number

On the Maximum Crossing Number

2018 ◽  
pp. 61-74
Author(s):  
Markus Chimani ◽  
Stefan Felsner ◽  
Stephen Kobourov ◽  
Torsten Ueckerdt ◽  
Pavel Valtr ◽  
...  
Keyword(s):  
Crossing Number
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