The average genus of oriented rational links with a given crossing number

The Genus Crossing Number

Pach and Tóth [6] introduced a new version of the crossing number parameter, called the degenerate crossing number, by considering proper drawings of a graph in the plane and counting multiple crossing of edges through the same point as a single crossing when all pairwise crossings of edges at that point are transversal. We propose a related parameter, called the genus crossing number, where ed...

Triple Crossing Number of 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 o...

On the Average Number of Rational Points on Curves of Genus 2

The second part of this conjecture is analogous to Conjecture 2.2 (i) in [PV], which considers hypersurfaces in P. If C is a curve of genus 2 as above and P = (a : y : b) is a rational point on C (i.e., we have F (a, b) = y with a, b coprime integers), then we denote by H(P ) the height H(a : b) = max{|a|, |b|} of its x-coordinate. Conjecture 2. Let ε > 0. Then there is a constant Bε and a Zari...

On the oriented chromatic number of graphs with given excess

The excess of a graph G is defined as the minimum number of edges that must be deleted from G in order to get a forest. We prove that every graph with excess at most k has chromatic number at most 1 2(3+ √ 1 + 8k) and that this bound is tight. Moreover, we prove that the oriented chromatic number of any graph with excess k is at most k+3, except for graphs having excess 1 and containing a direc...

Genus, Treewidth, and Local Crossing Number