Narrow Results

We consider a random collection of n line segments, in which the segments are independently drawn by picking the midpoint according to a given distribution in the plane, by picking a uniformly distributed random direction, and by adding an independently sampled segment length. We study the expected behavior of the number of intersections as a function of n, the lengths of the line segments, and the distribution of the midpoints. This includes a universally applicable law of large numbers.

Given a knot in the 3-sphere, the non-orientable 4-genus or 4-dimensional crosscap number of a knot is the minimal first Betti number of non-orientable surfaces, smoothly and properly embedded in the 4-ball, with boundary the knot. In this paper, we calculate the non-orientable 4-genus of knots with crossing number 10.

Studying algebraic structures via graphs and hypergraphs assigned to them can be of interest. Especially, computing the genus of a graph as a topological index leads to a better understanding of the related algebraic structure. In this direction we apply a hypergraph, namely $3$-zero divisor hypergraph assigned to a commutative ring and study its genus. In this paper, we characterize all finite commutative nonlocal rings $A$ with identity whose ${{\mathscr{H}}}_3(A)$ has genus two. Further, we classify all finite commutative nonlocal rings $A$ whose ${{\mathscr{H}}}_3(A)$ has crosscap two. Moreover, we provide a MATLAB code for calculating $3$-zero-divisor of ${\mathbb{Z}}_n$ and the hyperedge of $3$-zero-divisor hypergraph of ${\mathbb{Z}}_n$.

In this paper, we investigate the crosscap of 3-annihilating-ideal hypergraph ${{𝒜}{𝒢}}_3(R)$ of a commutative ring $R$ and the topological embedding of ${{𝒜}{𝒢}}_3(R)$ to the nonorientable compact surfaces. Furthermore, we determine all Artinian commutative non-local rings R (up to isomorphism) such that ${{𝒜}{𝒢}}_3(R)$ is a projective graph.