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

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 General Lower Bounds for the Minor Crossing Number of Graphs

2010 ◽  
Vol 44 (2) ◽  
pp. 463-483 ◽  
Author(s):  
Drago Bokal ◽  
Éva Czabarka ◽  
László A. Székely ◽  
Imrich Vrt’o
Keyword(s):  
Lower Bounds ◽  
Crossing Number

scholarly journals COMPLEXITY OF LINKS IN 3-MANIFOLDS

2009 ◽  
Vol 18 (10) ◽  
pp. 1439-1458 ◽  
Author(s):  
EKATERINA PERVOVA ◽  
CARLO PETRONIO
Keyword(s):  
Lower Bounds ◽  
Crossing Number ◽  
Upper And Lower Bounds ◽  
Recent Theory ◽  
Asymptotic Estimates ◽  
Connected Sum ◽  
Hyperbolic Volume ◽  
Do So

We introduce a complexity c(X) ∈ ℕ for pairs X = (M,L), where M is a closed orientable 3-manifold and L ⊂ M is a link. The definition employs simple spines, but for well-behaved X 's , we show that c(X) equals the minimal number of tetrahedra in a triangulation of M containing L in its 1-skeleton. Slightly adapting Matveev's recent theory of roots for graphs, we carefully analyze the behaviour of c under connected sum away from and along the link. We show in particular that c is almost always additive, describing in detail the circumstances under which it is not. To do so we introduce a certain (0, 2)-root for a pair X, we show that it is well-defined, and we prove that X has the same complexity as its (0, 2)-root. We then consider, for links in the 3-sphere, the relations of c with the crossing number and with the hyperbolic volume of the exterior, establishing various upper and lower bounds. We also specialize our analysis to certain infinite families of links, providing rather accurate asymptotic estimates.

scholarly journals New Lower Bounds for the Number of (≤ k)-Edges and the Rectilinear Crossing Number of Kn

2007 ◽  
Vol 38 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Oswin Aichholzer ◽  
Jesus Garcia ◽  
David Orden ◽  
Pedro Ramos
Keyword(s):  
Lower Bounds ◽  
Crossing Number

scholarly journals On the Poisson Integral of Step Functions and Minimal Surfaces

2002 ◽  
Vol 45 (1) ◽  
pp. 154-160 ◽  
Author(s):  
Allen Weitsman
Keyword(s):  
Minimal Surface ◽  
Lower Bounds ◽  
Minimal Surfaces ◽  
Step Function ◽  
Poisson Integral ◽  
Harmonic Mappings ◽  
Step Functions ◽  
Number Of Zeros ◽  
Univalent Harmonic Mappings

AbstractApplications of minimal surface methods are made to obtain information about univalent harmonic mappings. In the case where the mapping arises as the Poisson integral of a step function, lower bounds for the number of zeros of the dilatation are obtained in terms of the geometry of the image.

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 Crossings, Colorings, and Cliques

10.37236/134 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Michael O. Albertson ◽  
Daniel W. Cranston ◽  
Jacob Fox
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Complete Graph ◽  
Crossing Number ◽  
Critical Graphs ◽  
Four Color Theorem

Albertson conjectured that if graph $G$ has chromatic number $r$, then the crossing number of $G$ is at least that of the complete graph $K_r$. This conjecture in the case $r=5$ is equivalent to the four color theorem. It was verified for $r=6$ by Oporowski and Zhao. In this paper, we prove the conjecture for $7 \leq r \leq 12$ using results of Dirac; Gallai; and Kostochka and Stiebitz that give lower bounds on the number of edges in critical graphs, together with lower bounds by Pach et al. on the crossing number of graphs in terms of the number of edges and vertices.

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 The minimum crossing number of essential tangles

2014 ◽  
Vol 23 (10) ◽  
pp. 1450054
Author(s):  
João Miguel Nogueira ◽  
António Salgueiro
Keyword(s):  
Lower Bounds ◽  
Crossing Number ◽  
Number Of Components

In this paper we compute the sharp lower bounds for the crossing number of n-string k-loop essential tangles. For essential tangles with only string components, we characterize the ones with the minimum crossing number for a given number of components, both when the tangle has knotted strings or only unknotted strings.

SINGULARITY KNOTS OF MINIMAL SURFACES IN ℝ4

2011 ◽  
Vol 20 (04) ◽  
pp. 513-546 ◽  
Author(s):  
MARC SORET ◽  
MARINA VILLE
Keyword(s):  
Minimal Surface ◽  
Branch Point ◽  
Minimal Surfaces ◽  
Crossing Number ◽  
Alexander Polynomial

We study knots in 𝕊3 obtained by the intersection of a minimal surface in ℝ4 with a small 3-sphere centered at a branch point. We construct new examples of minimal knots. In particular we show the existence of non-fibered minimal knots. We show that simple minimal knots are either reversible or fully amphicheiral; this yields an obstruction for a given knot to be a simple minimal knot. Properties and invariants of these knots such as the algebraic crossing number of a braid representative and the Alexander polynomial are studied.

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