ON THE ADDITIVITY OF CROSSING NUMBERS OF GRAPHS

2008 ◽  
Vol 17 (09) ◽  
pp. 1043-1050 ◽  
Author(s):  
JESÚS LEAÑOS ◽  
GELASIO SALAZAR
Keyword(s):  
Connected Graph ◽  
Knot Theory ◽  
Crossing Number ◽  
Connected Sum ◽  
Critical Graphs ◽  
Crossing Numbers ◽  
The Individual

We describe a relationship between the crossing number of a graph G with a 2-edge-cut C and the crossing numbers of the components of G-C. Let G be a connected graph with a 2-edge-cut C := [V1,V2]. Let u1u2, v1v2 be the edges of C, so that ui,vi ∈ Vi for i = 1,2, and let Gi := G[Vi] and G'i := Gi + uivi. We show that if either G1 or G2 is not connected, then cr (G) = cr (G1) + cr (G2), and that if they are both connected then cr (G) = cr (G'1) + cr (G'2). We use this to show how to decompose crossing-critical graphs with 2-edge-cuts into smaller, 3-edge-connected crossing-critical graphs. We also observe that this settles a question arising from knot theory, raised by Sawollek, by describing exactly under which conditions the crossing number of the connected sum of two graphs equals the sum of the crossing numbers of the individual graphs.

THE ADDITIVITY OF CROSSING NUMBERS

2004 ◽  
Vol 13 (07) ◽  
pp. 857-866 ◽  
Author(s):  
YUANAN DIAO
Keyword(s):  
Knot Theory ◽  
Difficult Problem ◽  
Crossing Number ◽  
The Other ◽  
Connected Sum ◽  
Torus Knots ◽  
Crossing Numbers ◽  
Weaker Conditions ◽  
Alternating Links ◽  
Alternating Link

It has long been conjectured that the crossing numbers of links are additive under the connected sum of links. This is a difficult problem in knot theory and has been open for more than 100 years. In fact, many questions of much weaker conditions are still open. For instance, it is not known whether Cr(K1#K2)≥Cr(K1) or Cr(K1#K2)≥Cr(K2) holds in general, here K1#K2 is the connected sum of K1 and K2 and Cr(K) stands for the crossing number of the link K. However, for alternating links K1 and K2, Cr(K1#K2)=Cr(K1)+Cr(K2) does hold. On the other hand, if K1 is an alternating link and K2 is any link, then we have Cr(K1#K2)≥Cr(K1). In this paper, we show that there exists a wide class of links over which the crossing number is additive under the connected sum operation. This class is different from the class of all alternating links. It includes all torus knots and many alternating links. Furthermore, if K1 is a connected sum of any given number of links from this class and K2 is a non-trivial knot, we prove that Cr(K1#K2)≥Cr(K1)+3.

On the Question of Genericity of Hyperbolic Knots

2018 ◽  
Vol 2020 (21) ◽  
pp. 7792-7828
Author(s):  
Andrei V Malyutin
Keyword(s):  
Knot Theory ◽  
Crossing Number ◽  
Connected Sum ◽  
Satellite Knot

Abstract A well-known conjecture in knot theory says that the proportion of hyperbolic knots among all of the prime knots of $n$ or fewer crossings approaches $1$ as $n$ approaches infinity. In this article, it is proved that this conjecture contradicts several other plausible conjectures, including the 120-year-old conjecture on additivity of the crossing number of knots under connected sum and the conjecture that the crossing number of a satellite knot is not less than that of its companion.

scholarly journals Lernaean knots and band surgery

2021 ◽  
Vol 33 (1) ◽  
pp. 23-46
Author(s):  
Yu. Belousov ◽  
M. Karev ◽  
A. Malyutin ◽  
A. Miller ◽  
E. Fominykh
Keyword(s):  
Knot Theory ◽  
Crossing Number ◽  
Connected Sum ◽  
Content Type

The paper is devoted to a line of the knot theory related to the conjecture on the additivity of the crossing number for knots under connected sum. A series of weak versions of this conjecture are proved. Many of these versions are formulated in terms of the band surgery graph also called the H ( 2 ) H(2) -Gordian graph.

DETERMINING CROSSING NUMBERS OF GRAPHS OF ORDER SIX USING CYCLIC PERMUTATIONS

2018 ◽  
Vol 98 (3) ◽  
pp. 353-362 ◽  
Author(s):  
MICHAL STAŠ
Keyword(s):  
Connected Graph ◽  
Crossing Number ◽  
Crossing Numbers

We extend known results concerning crossing numbers by giving the crossing number of the join product$G+D_{n}$, where the connected graph$G$consists of one$4$-cycle and of two leaves incident with the same vertex of the$4$-cycle, and$D_{n}$consists of$n$isolated vertices. The proofs are done with the help of software that generates all cyclic permutations for a given number$k$and creates a graph for calculating the distances between all$(k-1)!$vertices of the graph.

scholarly journals On the Crossing Numbers of the Joining of a Specific Graph on Six Vertices with the Discrete Graph

Symmetry ◽  
2020 ◽  
Vol 12 (1) ◽  
pp. 135
Author(s):  
Michal Staš
Keyword(s):  
Crossing Number ◽  
Crossing Numbers ◽  
The Individual ◽  
Discrete Graph

In the paper, we extend known results concerning crossing numbers of join products of small graphs of order six with discrete graphs. The crossing number of the join product G ∗ + D n for the graph G ∗ on six vertices consists of one vertex which is adjacent with three non-consecutive vertices of the 5-cycle. The proofs were based on the idea of establishing minimum values of crossings between two different subgraphs that cross the edges of the graph G ∗ exactly once. These minimum symmetrical values are described in the individual symmetric tables.

scholarly journals Determining crossing numbers of the join products of two specific graphs of order six with the discrete graph

Filomat ◽  
2020 ◽  
Vol 34 (9) ◽  
pp. 2829-2846
Author(s):  
Michal Stas
Keyword(s):  
Connected Graph ◽  
Crossing Number ◽  
Topological Approach ◽  
Crossing Numbers ◽  
Minimum Numbers ◽  
Discrete Graph

The main aim of the paper is to give the crossing number of the join product G* + Dn for the connected graph G* of order six consisting of P4 + D1 and of one leaf incident with some inner vertex of the path P4 on four vertices, and where Dn consists of n isolated vertices. In the proofs, it will be extend the idea of the minimum numbers of crossings between two different subgraphs from the set of subgraphs which do not cross the edges of the graph G* onto the set of subgraphs by which the edges of G* are crossed exactly once. Due to the mentioned algebraic topological approach, we are able to extend known results concerning crossing numbers for join products of new graphs. The proofs are 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 (k-1)! vertices of the graph. Finally, by adding one edge to the graph G*, we are able to obtain the crossing number of the join product of one other graph with the discrete graph Dn.

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 The Crossing Numbers of Join Products of Paths and Cycles with Four Graphs of Order Five

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1277
Author(s):  
Michal Staš
Keyword(s):  
Connected Graph ◽  
Crossing Numbers ◽  
Tripartite Graph ◽  
Complete Tripartite Graph

The main aim of the paper is to establish the crossing numbers of the join products of the paths and the cycles on n vertices with a connected graph on five vertices isomorphic to the graph K1,1,3\e obtained by removing one edge e incident with some vertex of order two from the complete tripartite graph K1,1,3. The proofs are done with the help of well-known exact values for the crossing numbers of the join products of subgraphs of the considered graph with paths and cycles. Finally, by adding some edges to the graph under consideration, we obtain the crossing numbers of the join products of other graphs with the paths and the cycles on n vertices.

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.

Totally knotted knots are prime

1982 ◽  
Vol 91 (3) ◽  
pp. 467-472
Author(s):  
J. C. Gomez-Larran¯aga
Keyword(s):  
Finite Number ◽  
Knot Theory ◽  
Piecewise Linear ◽  
Connected Sum ◽  
Higher Dimensional ◽  
The Family ◽  
Linear Category

Throughout, the word knot means a subspace of the 3-sphere S3 homeomorphic with the 1-sphere S1. Any knot can be expressed as a connected sum of a finite number of prime knots in a unique way (13), we consider the trivial knot a non-prime knot. (For higher dimensional knots, factorization and uniqueness have been studied in (1).) However given a knot it is difficult to determine if it is prime or not. We prove that totally knotted knots, see definition in §2, are prime in theorem 1, give a class of examples in theorem 2 and investigate how the last result can be applied to the conjecture that the family Y of unknotting number one knots are prime. (See problem 2 in (5).) At the end, prime tangles as defined by W. B. R. Lickerish are used to prove that in a certain family of knots, related somewhat to Y, there is just one non-prime knot: the square knot. The paper should be interpreted as being in the piecewise linear category. Standard definitions of 3-manifolds and knot theory may be found in (6) and (11) respectively.

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