scholarly journals Coherent band pathways between knots and links

2015 ◽  
Vol 24 (02) ◽  
pp. 1550006 ◽  
Author(s):  
Dorothy Buck ◽  
Kai Ishihara
Keyword(s):  
Lower Bounds ◽  
Recombinant Dna ◽  
Crossing Number ◽  
Dna Molecules ◽  
Minimum Number ◽  
Knots And Links

We categorize coherent band (aka nullification) pathways between knots and 2-component links. Additionally, we characterize the minimal coherent band pathways (with intermediates) between any two knots or 2-component links with small crossing number. We demonstrate these band surgeries for knots and links with small crossing number. We apply these results to place lower bounds on the minimum number of recombinant events separating DNA configurations, restrict the recombination pathways and determine chirality and/or orientation of the resulting recombinant DNA molecules.

scholarly journals TRIPLE CROSSING NUMBER OF KNOTS AND LINKS

2013 ◽  
Vol 22 (02) ◽  
pp. 1350006 ◽  
Author(s):  
COLIN ADAMS
Keyword(s):  
Lower Bounds ◽  
Crossing Number ◽  
Upper And Lower Bounds ◽  
Minimum Number ◽  
Bracket Polynomial ◽  
Knots And Links

A triple crossing is a crossing in a projection of a knot or link that has three strands of the knot passing straight through it. A triple crossing projection is a projection such that all of the crossings are triple crossings. We prove that every knot and link has a triple crossing projection and then investigate c3(K), which is the minimum number of triple crossings in a projection of K. We obtain upper and lower bounds on c3(K) in terms of the traditional crossing number and show that both are realized. We also relate triple crossing number to the span of the bracket polynomial and use this to determine c3(K) for a variety of knots and links. We then use c3(K) to obtain bounds on the volume of a hyperbolic knot or link. We also consider extensions to cn(K).

scholarly journals On the Crossing Numbers of the Join Products of Six Graphs of Order Six with Paths and Cycles

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2441
Author(s):  
Michal Staš
Keyword(s):  
Lower Bounds ◽  
Crossing Number ◽  
Crossing Numbers ◽  
Symmetric Graphs ◽  
The Family ◽  
Minimum Number ◽  
The Common ◽  
First Time ◽  
Edge Crossings

The crossing number of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main purpose of this paper is to determine the crossing numbers of the join products of six symmetric graphs on six vertices with paths and cycles on n vertices. The idea of configurations is generalized for the first time onto the family of subgraphs whose edges cross the edges of the considered graph at most once, and their lower bounds of necessary numbers of crossings are presented in the common symmetric table. Some proofs of the join products with cycles are done with the help of several well-known auxiliary statements, the idea of which is extended by a suitable classification of subgraphs that do not cross the edges of the examined graphs.

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 GENETIC HETEROZYGOSITY IN UNREPLICATED BACTERIOPHAGE Λ RECOMBINANTS

Genetics ◽  
1975 ◽  
Vol 81 (1) ◽  
pp. 33-50
Author(s):  
Raymond L White ◽  
Maurice S Fox
Keyword(s):  
Dna Synthesis ◽  
Base Pair ◽  
Recombinant Dna ◽  
Dna Molecules ◽  
Bacteriophage Λ ◽  
Negative Interference ◽  
Genetic Interval ◽  
Parental Material ◽  
Genetic Heterozygosity ◽  
Genetic Contributions

ABSTRACT Bacteriophage crosses using density-labeled parents have been carried out under conditions restricting DNA synthesis. The parental material and genetic contributions to progeny manifesting recombination within a genetic interval sufficiently short to exhibit high negative interference have been examined. The unreplicated products of recombination isolated as phage particles appear to contain long continuous heteroduplex regions which are heterozygous for the closely linked markers. Recombination between closely linked markers seems to be the consequence of the removal of base-pair mismatches that are present within the heteroduplex regions. This localized reduction of heterozygosity within the heteroduplex regions that join the parental components of recombinant DNA molecules can account for high negative interference.

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].

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
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