scholarly journals Multi-crossing number for knots and the Kauffman bracket polynomial

2016 ◽  
Vol 164 (1) ◽  
pp. 147-178 ◽  
Author(s):  
COLIN ADAMS ◽  
ORSOLA CAPOVILLA-SEARLE ◽  
JESSE FREEMAN ◽  
DANIEL IRVINE ◽  
SAMANTHA PETTI ◽  
...  
Keyword(s):  
Singular Point ◽  
Lower Bound ◽  
Crossing Number ◽  
Crossing Numbers ◽  
Kauffman Bracket ◽  
Classic Result ◽  
Formula Group ◽  
Image Position ◽  
Bracket Polynomial ◽  
Extensive List

AbstractA multi-crossing (or n-crossing) is a singular point in a projection of a knot or link at which n strands cross so that each strand bisects the crossing. We generalise the classic result of Kauffman, Murasugi and Thistlethwaite relating the span of the bracket polynomial to the double-crossing number of a link, span〈K〉 ⩽ 4c2, to the n-crossing number. We find the following lower bound on the n-crossing number in terms of the span of the bracket polynomial for any n ⩾ 3: $$\text{span} \langle K \rangle \leq \left(\left\lfloor\frac{n^2}{2}\right\rfloor + 4n - 8\right) c_n(K).$$ We also explore n-crossing additivity under composition, and find that for n ⩾ 4 there are examples of knots K1 and K2 such that cn(K1#K2) = cn(K1) + cn(K2) − 1. Further, we present the the first extensive list of calculations of n-crossing numbers of knots. Finally, we explore the monotonicity of the sequence of n-crossings of a knot, which we call the crossing spectrum.

scholarly journals Minimum Number of Monotone Subsequences of Length 4 in Permutations

2014 ◽  
Vol 24 (4) ◽  
pp. 658-679 ◽  
Author(s):  
JÓZSEF BALOGH ◽  
PING HU ◽  
BERNARD LIDICKÝ ◽  
OLEG PIKHURKO ◽  
BALÁZS UDVARI ◽  
...  
Keyword(s):  
Lower Bound ◽  
Complete Graphs ◽  
Formula Group ◽  
Minimum Number ◽  
Additional Stability ◽  
Edge Colourings ◽  
Image Position

We show that for every sufficiently largen, the number of monotone subsequences of length four in a permutation onnpoints is at least\begin{equation*} \binom{\lfloor{n/3}\rfloor}{4} + \binom{\lfloor{(n+1)/3}\rfloor}{4} + \binom{\lfloor{(n+2)/3}\rfloor}{4}. \end{equation*}Furthermore, we characterize all permutations on [n] that attain this lower bound. The proof uses the flag algebra framework together with some additional stability arguments. This problem is equivalent to some specific type of edge colourings of complete graphs with two colours, where the number of monochromaticK4is minimized. We show that all the extremal colourings must contain monochromaticK4only in one of the two colours. This translates back to permutations, where all the monotone subsequences of length four are all either increasing, or decreasing only.

scholarly journals Braid index bounds ropelength from below

2020 ◽  
Vol 29 (04) ◽  
pp. 2050019
Author(s):  
Yuanan Diao
Keyword(s):  
Lower Bound ◽  
General Formula ◽  
Crossing Number ◽  
Crossing Numbers ◽  
Braid Index ◽  
Alternating Links

For an unoriented link [Formula: see text], let [Formula: see text] be the ropelength of [Formula: see text]. It is known that in general [Formula: see text] is at least of the order [Formula: see text], and at most of the order [Formula: see text] where [Formula: see text] is the minimum crossing number of [Formula: see text]. Furthermore, it is known that there exist families of (infinitely many) links with the property [Formula: see text]. A long standing open conjecture states that if [Formula: see text] is alternating, then [Formula: see text] is at least of the order [Formula: see text]. In this paper, we show that the braid index of a link also gives a lower bound of its ropelength. More specifically, we show that there exists a constant [Formula: see text] such that [Formula: see text] for any [Formula: see text], where [Formula: see text] is the largest braid index among all braid indices corresponding to all possible orientation assignments of the components of [Formula: see text] (called the maximum braid index of [Formula: see text]). Consequently, [Formula: see text] for any link [Formula: see text] whose maximum braid index is proportional to its crossing number. In the case of alternating links, the maximum braid indices for many of them are proportional to their crossing numbers hence the above conjecture holds for these alternating links.

scholarly journals Crossing Numbers and Hard Erdős Problems in Discrete Geometry

1997 ◽  
Vol 6 (3) ◽  
pp. 353-358 ◽  
Author(s):  
LÁSZLÓ A. SZÉKELY
Keyword(s):  
Lower Bound ◽  
Discrete Geometry ◽  
Crossing Number ◽  
Plane Geometry ◽  
Crossing Numbers ◽  
Minimum Number

We show that an old but not well-known lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the number of incidences among points and lines, the maximum number of unit distances among n points, the minimum number of distinct distances among n points.

Mass transference principle for limsup sets generated by rectangles

2015 ◽  
Vol 158 (3) ◽  
pp. 419-437 ◽  
Author(s):  
BAO-WEI WANG ◽  
JUN WU ◽  
JIAN XU
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Hausdorff Measure ◽  
Unit Cube ◽  
Transference Principle ◽  
Theoretical Statement ◽  
Formula Group ◽  
Image Position ◽  
Full Lebesgue Measure

AbstractWe generalise the mass transference principle established by Beresnevich and Velani to limsup sets generated by rectangles. More precisely, let {xn}n⩾1 be a sequence of points in the unit cube [0, 1]d with d ⩾ 1 and {rn}n⩾1 a sequence of positive numbers tending to zero. Under the assumption of full Lebesgue measure theoretical statement of the set \begin{equation*}\big\{x\in [0,1]^d: x\in B(x_n,r_n), \ {{\rm for}\, {\rm infinitely}\, {\rm many}}\ n\in \mathbb{N}\big\},\end{equation*} we determine the lower bound of the Hausdorff dimension and Hausdorff measure of the set \begin{equation*}\big\{x\in [0,1]^d: x\in B^{a}(x_n,r_n), \ {{\rm for}\, {\rm infinitely}\, {\rm many}}\ n\in \mathbb{N}\big\},\end{equation*} where a = (a1, . . ., ad) with 1 ⩽ a1 ⩽ a2 ⩽ . . . ⩽ ad and Ba(x, r) denotes a rectangle with center x and side-length (ra1, ra2,. . .,rad). When a1 = a2 = . . . = ad, the result is included in the setting considered by Beresnevich and Velani.

scholarly journals Normal Numbers and the Normality Measure

2013 ◽  
Vol 22 (3) ◽  
pp. 342-345 ◽  
Author(s):  
CHRISTOPH AISTLEITNER
Keyword(s):  
Lower Bound ◽  
Binary Sequence ◽  
Normal Numbers ◽  
Small Discrepancy ◽  
Normal Number ◽  
Formula Group ◽  
Image Position ◽  
Normality Measure

In a paper published in this journal, Alon, Kohayakawa, Mauduit, Moreira and Rödl proved that the minimal possible value of the normality measure of an N-element binary sequence satisfies \begin{equation*} \biggl( \frac{1}{2} + o(1) \biggr) \log_2 N \leq \min_{E_N \in \{0,1\}^N} \mathcal{N}(E_N) \leq 3 N^{1/3} (\log N)^{2/3} \end{equation*} for sufficiently large N, and conjectured that the lower bound can be improved to some power of N. In this note it is observed that a construction of Levin of a normal number having small discrepancy gives a construction of a binary sequence EN with (EN) = O((log N)2), thus disproving the conjecture above.

Space Crossing Numbers

2012 ◽  
Vol 21 (3) ◽  
pp. 358-373 ◽  
Author(s):  
BORIS BUKH ◽  
ALFREDO HUBARD
Keyword(s):  
Lower Bound ◽  
Random Graphs ◽  
Crossing Number ◽  
Sharp Bounds ◽  
Crossing Numbers ◽  
Crossing Lemma ◽  
Rectilinear Space

We define a variant of the crossing number for an embedding of a graphGinto ℝ3, and prove a lower bound on it which almost implies the classical crossing lemma. We also give sharp bounds on the rectilinear space crossing numbers of pseudo-random graphs.

scholarly journals Saturated Subgraphs of the Hypercube

2016 ◽  
Vol 26 (1) ◽  
pp. 52-67 ◽  
Author(s):  
J. ROBERT JOHNSON ◽  
TREVOR PINTO
Keyword(s):  
Lower Bound ◽  
Constant Factor ◽  
Formula Group ◽  
Minimum Number ◽  
Image Position

We say a graph is (Qn,Qm)-saturatedif it is a maximalQm-free subgraph of then-dimensional hypercubeQn. A graph is said to be (Qn,Qm)-semi-saturatedif it is a subgraph ofQnand adding any edge forms a new copy ofQm. The minimum number of edges a (Qn,Qm)-saturated graph (respectively (Qn,Qm)-semi-saturated graph) can have is denoted by sat(Qn,Qm) (respectivelys-sat(Qn,Qm)). We prove that$$ \begin{linenomath} \lim_{n\to\infty}\ffrac{\sat(Q_n,Q_m)}{e(Q_n)}=0, \end{linenomath}$$for fixedm, disproving a conjecture of Santolupo that, whenm=2, this limit is 1/4. Further, we show by a different method that sat(Qn,Q2)=O(2n), and thats-sat(Qn,Qm)=O(2n), for fixedm. We also prove the lower bound$$ \begin{linenomath} \ssat(Q_n,Q_m)\geq \ffrac{m+1}{2}\cdot 2^n, \end{linenomath}$$thus determining sat(Qn,Q2) to within a constant factor, and discuss some further questions.

Colourings of Uniform Hypergraphs with Large Girth and Applications

2017 ◽  
Vol 27 (2) ◽  
pp. 245-273 ◽  
Author(s):  
ANDREY KUPAVSKII ◽  
DMITRY SHABANOV
Keyword(s):  
Lower Bound ◽  
Arithmetic Progression ◽  
Absolute Constant ◽  
Combinatorial Problem ◽  
Uniform Hypergraphs ◽  
Probabilistic Technique ◽  
Formula Group ◽  
Large Girth ◽  
Image Position

This paper deals with a combinatorial problem concerning colourings of uniform hypergraphs with large girth. We prove that ifHis ann-uniform non-r-colourable simple hypergraph then its maximum edge degree Δ(H) satisfies the inequality$$ \Delta(H)\geqslant c\cdot r^{n-1}\ffrac{n(\ln\ln n)^2}{\ln n} $$for some absolute constantc> 0.As an application of our probabilistic technique we establish a lower bound for the classical van der Waerden numberW(n, r), the minimum naturalNsuch that in an arbitrary colouring of the set of integers {1,. . .,N} withrcolours there exists a monochromatic arithmetic progression of lengthn. We prove that$$ W(n,r)\geqslant c\cdot r^{n-1}\ffrac{(\ln\ln n)^2}{\ln n}. $$

scholarly journals Outerplanar Crossing Numbers, the Circular Arrangement Problem and Isoperimetric Functions

10.37236/1834 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Éva Czabarka ◽  
Ondrej Sýkora ◽  
László A. Székely ◽  
Imrich Vrťo
Keyword(s):  
Lower Bound ◽  
General Setting ◽  
Crossing Number ◽  
Linear Arrangement ◽  
Crossing Numbers ◽  
Arrangement Problem ◽  
Isoperimetric Functions ◽  
Circular Arrangement ◽  
Graph Families ◽  
Linear Arrangement Problem

We extend a lower bound due to Shahrokhi, Sýkora, Székely and Vrťo for the outerplanar crossing number (in other terminologies also called convex, circular and one-page book crossing number) to a more general setting. In this setting we can show a better lower bound for the outerplanar crossing number of hypercubes than the best lower bound for the planar crossing number. We exhibit further sequences of graphs, whose outerplanar crossing number exceeds by a factor of $\log n$ the planar crossing number of the graph. We study the circular arrangement problem, as a lower bound for the linear arrangement problem, in a general fashion. We obtain new lower bounds for the circular arrangement problem. All the results depend on establishing good isoperimetric functions for certain classes of graphs. For several graph families new near-tight isoperimetric functions are established.

scholarly journals Erdős–Ko–Rado for Random Hypergraphs: Asymptotics and Stability

2017 ◽  
Vol 26 (3) ◽  
pp. 406-422
Author(s):  
MARCELO M. GAUY ◽  
HIÊP HÀN ◽  
IGOR C. OLIVEIRA
Keyword(s):  
Lower Bound ◽  
Uniform Hypergraph ◽  
Small Interval ◽  
Asymptotic Size ◽  
Intersecting Family ◽  
Formula Group ◽  
Image Position ◽  
Random Hypergraphs

We investigate the asymptotic version of the Erdős–Ko–Rado theorem for the random k-uniform hypergraph $\mathcal{H}$k(n, p). For 2⩽k(n) ⩽ n/2, let $N=\binom{n}k$ and $D=\binom{n-k}k$. We show that with probability tending to 1 as n → ∞, the largest intersecting subhypergraph of $\mathcal{H}$ has size $$(1+o(1))p\ffrac kn N$$ for any $$p\gg \ffrac nk\ln^2\biggl(\ffrac nk\biggr)D^{-1}.$$ This lower bound on p is asymptotically best possible for k = Θ(n). For this range of k and p, we are able to show stability as well.A different behaviour occurs when k = o(n). In this case, the lower bound on p is almost optimal. Further, for the small interval D−1 ≪ p ⩽ (n/k)1−ϵD−1, the largest intersecting subhypergraph of $\mathcal{H}$k(n, p) has size Θ(ln(pD)ND−1), provided that $k \gg \sqrt{n \ln n}$.Together with previous work of Balogh, Bohman and Mubayi, these results settle the asymptotic size of the largest intersecting family in $\mathcal{H}$k, for essentially all values of p and k.

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