arXiv subjects

Chaim Even-Zohar

Publications and source records attributed to Chaim Even-Zohar.

25 records · Page 2Linked to original sources

arXiv2016

Invariants of Random Knots and Links

We study random knots and links in R^3 using the Petaluma model, which is based on the petal projections developed by Adams et al. (2012). In this model we obtain a formula for the distribution of the linking number of a random two-component link. We also obtain formulas for the expectations and the higher moments of the Casson invariant and the order-3 knot invariant v3. These are the first precise formulas given for the distributions of invariants in any model for random knots or links. We also use numerical computation to compare these to other random knot and link models, such as those based on grid diagrams.

math.GT↗

arXiv2016

Crossing Numbers of Random Two-Bridge Knots

In a previous work, the first and third authors studied a random knot model for all two-bridge knots using billiard table diagrams. Here we present a closed formula for the distribution of the crossing numbers of such random knots. We also show that the probability of any given knot appearing in this model decays to zero at an exponential rate as the length of the billiard table goes to infinity. This confirms a conjecture from the previous work.

math.GT↗

arXiv2014

A Note on the Inducibility of 4-vertex Graphs

There is much recent interest in understanding the density at which constant size graphs can appear in a very large graph. Specifically, the inducibility of a graph H is its extremal density, as an induced subgraph of G, where |G| -> infinity. Already for 4-vertex graphs many questions are still open. Thus, the inducibility of the 4-path was addressed in a construction of Exoo (1986), but remains unknown. Refuting a conjecture of Erdos, Thomason (1997) constructed graphs with a small density of both 4-cliques and 4-anticliques. In this note, we merge these two approaches and construct better graphs for both problems.

math.CO↗

arXiv2014

Triply Existentially Complete Triangle-Free Graphs

A triangle-free graph G is called k-existentially complete if for every induced k-vertex subgraph H of G, every extension of H to a (k+1)-vertex triangle-free graph can be realized by adding another vertex of G to H. Cherlin asked whether k-existentially complete triangle-free graphs exist for every k. Here we present known and new constructions of 3-existentially complete triangle-free graphs.

math.CO↗

arXiv2014

The Freiman--Ruzsa Theorem over Finite Fields

Let G be a finite abelian group of torsion r and let A be a subset of G. The Freiman--Ruzsa theorem asserts that if |A+A| < K|A| then A is contained in a coset of a subgroup of G of size at most r^{K^4}K^2|A|. It was conjectured by Ruzsa that the subgroup size can be reduced to r^{CK}|A| for some absolute constant C >= 2. This conjecture was verified for r = 2 in a sequence of recent works, which have, in fact, yielded a tight bound. In this work, we establish the same conjecture for any prime torsion.

math.CO↗

arXiv2012

On Sums of Generating Sets in (Z_2)^n

Let A and B be two affinely generating sets of (Z_2)^n. As usual, we denote their Minkowski sum by A+B. How small can A+B be, given the cardinalities of A and B? We give a tight answer to this question. Our bound is attained when both A and B are unions of cosets of a certain subgroup of (Z_2)^n. These cosets are arranged as Hamming balls, the smaller of which has radius 1. By similar methods, we re-prove the Freiman-Ruzsa theorem in (Z_2)^n, with an optimal upper bound. Denote by F(K) the maximal spanning constant | |/|A|, over all subsets A of (Z_2)^n with doubling constant |A+A|/|A| < K. We explicitly calculate F(K), and in particular show that 4^K / 4K < F(K) (1+o(1)) < 4^K / 2K. This improves the estimate F(K) = poly(K) 4^K, found recently by Green and Tao and by Konyagin.

math.CO↗

arXiv2011

Small asymmetric sumsets in elementary abelian 2-groups

Let A and B be subsets of an elementary abelian 2-group G, none of which are contained in a coset of a proper subgroup. Extending onto potentially distinct summands a result of Hennecart and Plagne, we show that if |A+B|<|A|+|B|, then either A+B=G, or the complement of A+B in G is contained in a coset of a subgroup of index at least 8, whence |A+B| is at least 7/8 |G|. We indicate conditions for the containment to be strict, and establish a refinement in the case where the sizes of A and B differ significantly.

math.CO↗