scholarly journals Minimum lattice length and ropelength of knots

2014 ◽  
Vol 23 (07) ◽  
pp. 1460009 ◽  
Author(s):  
Kyungpyo Hong ◽  
Hyoungjun Kim ◽  
Seungsang Oh ◽  
Sungjong No
Keyword(s):  
Upper Bounds ◽  
Crossing Number ◽  
Minimum Length ◽  
Cubic Lattice ◽  
Normal Tube

Let Len (K) be the minimum length of a knot on the cubic lattice (namely the minimum length necessary to construct the knot in the cubic lattice). This paper provides upper bounds for Len (K) of a nontrivial knot K in terms of its crossing number c(K) as follows: [Formula: see text] The ropelength of a knot is the quotient of its length by its thickness, the radius of the largest embedded normal tube around the knot. We also provide upper bounds for the minimum ropelength Rop (K) which is close to twice Len (K): [Formula: see text]

Upper bounds on the minimum length of cubic lattice knots

2013 ◽  
Vol 46 (12) ◽  
pp. 125001 ◽  
Author(s):  
Kyungpyo Hong ◽  
Sungjong No ◽  
Seungsang Oh
Keyword(s):  
Upper Bounds ◽  
Minimum Length ◽  
Cubic Lattice

Idempotent depth in semigroups of order-preserving mappings

1994 ◽  
Vol 124 (5) ◽  
pp. 1045-1058 ◽  
Author(s):  
Peter M. Higgins
Keyword(s):  
Upper Bounds ◽  
Minimum Length ◽  
Finite Chain

We introduce algorithms for calculating minimum length factorisations of order-preserving mappings on a finite chain into products of idempotents, and into products of idempotents of defect one. The least upper bounds for these lengths are given.

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 Upper bounds for ropelength as a function of crossing number

2004 ◽  
Vol 135 (1-3) ◽  
pp. 253-264 ◽  
Author(s):  
Jason Cantarella ◽  
X.W.C Faber ◽  
Chad A Mullikin
Keyword(s):  
Upper Bounds ◽  
Crossing Number

Upper Bounds for the Connective Constant of Self-Avoiding Walks

1993 ◽  
Vol 2 (2) ◽  
pp. 115-136 ◽  
Author(s):  
Sven Erick Alm
Keyword(s):  
Large Class ◽  
Triangular Lattice ◽  
Upper Bounds ◽  
Square Lattice ◽  
Numerical Application ◽  
Cubic Lattice ◽  
Simple Cubic ◽  
Simple Cubic Lattice ◽  
Connective Constant ◽  
Self Avoiding Walks

We present a method for obtaining upper bounds for the connective constant of self-avoiding walks. The method works for a large class of lattices, including all that have been studied in connection with self-avoiding walks. The bound is obtained as the largest eigenvalue of a certain matrix. Numerical application of the method has given improved bounds for all lattices studied, e.g. μ < 2.696 for the square lattice, μ < 4.278 for the triangular lattice and μ < 4.756 for the simple cubic lattice.

scholarly journals On measures of nonplanarity of cubic graphs

2018 ◽  
Vol 11 (2) ◽  
Author(s):  
Leonid Plachta
Keyword(s):  
Upper Bounds ◽  
Crossing Number ◽  
Cubic Graphs ◽  
Minimum Order ◽  
Edge Deletion ◽  
Connected Sum ◽  
Minimal Genus ◽  
Two Measures

We study two measures of nonplanarity of cubic graphs G, the genus γ (G), and the edge deletion number ed(G). For cubic graphs of small orders these parameters are compared with another measure of nonplanarity, the rectilinear crossing number (G). We introduce operations of connected sum, specified for cubic graphs G, and show that under certain conditions the parameters γ(G) and ed(G) are additive (subadditive) with respect to them.The minimal genus graphs (i.e. the cubic graphs of minimum order with given value of genus γ) and the minimal edge deletion graphs (i.e. cubic graphs of minimum order with given value of edge deletion number ed) are introduced and studied. We provide upper bounds for the order of minimal genus and minimal edge deletion graphs.

scholarly journals Crosscap number of knots and volume bounds

2020 ◽  
Vol 31 (13) ◽  
pp. 2050111
Author(s):  
Noboru Ito ◽  
Yusuke Takimura
Keyword(s):  
Jones Polynomial ◽  
Upper Bounds ◽  
Crossing Number ◽  
Proved Theorem ◽  
Knot Invariants ◽  
Chord Diagrams ◽  
Hyperbolic Volume ◽  
Knot Invariant ◽  
Twist Number

In this paper, we obtain the crosscap number of any alternating knots by using our recently-introduced diagrammatic knot invariant (Theorem 1). The proof is given by properties of chord diagrams (Kindred proved Theorem 1 independently via other techniques). For non-alternating knots, we give Theorem 2 that generalizes Theorem 1. We also improve known formulas to obtain upper bounds of the crosscap number of knots (alternating or non-alternating) (Theorem 3). As a corollary, this paper connects crosscap numbers and our invariant with other knot invariants such as the Jones polynomial, twist number, crossing number, and hyperbolic volume (Corollaries 1–7). In Appendix A, using Theorem 1, we complete giving the crosscap numbers of the alternating knots with up to 11 crossings including those of the previously unknown values for [Formula: see text] knots (Tables A.1).

SYNCHRONIZING QUASI-EULERIAN AND QUASI-ONE-CLUSTER AUTOMATA

2013 ◽  
Vol 24 (06) ◽  
pp. 729-745 ◽  
Author(s):  
MIKHAIL V. BERLINKOV
Keyword(s):  
Markov Chains ◽  
Recent Result ◽  
Upper Bounds ◽  
Minimum Length ◽  
Extension Method ◽  
Synchronizing Automata

We describe a new version of the so-called extension method that was used to prove quadratic upper bounds on the minimum length of reset words for various important classes of synchronizing automata. Our approach is formulated in terms of Markov chains; it is in a sense dual to the usual extension method and improves on a recent result by Jungers. As an application, we obtain a quadratic upper bounds on the minimum length of reset words for generalizations of Eulerian and one-cluster automata. Finally, we show that the proposed approach is in some sense equivalent to the extension method.

scholarly journals Equilateral stick number of knots

2014 ◽  
Vol 23 (07) ◽  
pp. 1460008 ◽  
Author(s):  
Hyoungjun Kim ◽  
Sungjong No ◽  
Seungsang Oh
Keyword(s):  
Upper Bound ◽  
American Mathematical Society ◽  
Upper Bounds ◽  
Crossing Number ◽  
Equal Length ◽  
Mathematical Society ◽  
Geometric Objects

An equilateral stick number s=(K) of a knot K is defined to be the minimal number of sticks required to construct a polygonal knot of K which consists of equal length sticks. Rawdon and Scharein [Upper bounds for equilateral stick numbers, in Physical Knots: Knotting, Linking, and Folding Geometric Objects in ℝ3, Contemporary Mathematics, Vol. 304 (American Mathematical Society, Providence, RI, 2002), pp. 55–76] found upper bounds for the equilateral stick numbers of all prime knots through 10 crossings by using algorithms in the software KnotPlot. In this paper, we find an upper bound on the equilateral stick number of a non-trivial knot K in terms of the minimal crossing number c(K) which is s=(K) ≤ 2c(K) + 2. Moreover if K is a non-alternating prime knot, then s=(K) ≤ 2c(K) - 2. Furthermore we find another upper bound on the equilateral stick number for composite knots which is s=(K1♯K2) ≤ 2c(K1) + 2c(K2).

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