Related papers: Self intersections on a pair of pants

We count the number of conics through two general points in complete intersections when this number is finite and give an application in terms of quasi-lines.

Consider a Jacobian elliptic surface $E \to C$ with a section $P$ of infinite order. Previous work of the first author and Urz\'ua over the complex numbers gives a bound on the number of tangencies between $P$ and a torsion section of $E$…

In this paper, we compute the number of self-intersections of a plane projection of a generic complete intersection curve defined by polynomials with the given support. Moreover, we discuss the tropical counterpart of this problem.

Let $\Sigma_{g,n}$ denote the compact orientable surface with genus $g\geq 2$ and boundary disjoint union of $n$ circles. By using a particular pants-decomposition of $\Sigma_{g,n},$ we obtain a formula that computes the Reidemeister…

High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to spherical and hyperbolic planes.…

We prove a symmetric version of B\'ezout's theorem. More precisely, we show that the symmetric orbit type of a transverse intersection of complex symmetric hypersurfaces in projective space is determined by the degrees. In the projective…

For suitable subgroups of a finitely generated group, we define the intersection number of one subgroup with another subgroup and show that this number is symmetric. We also give an interpretation of this number.

Our main theorem asserts that every Farey graph embedded in the 1-skeleton of the pants complex of any finite type surface is totally geodesic.

Consider two paths $\phi,\psi:[0;1]\to [0;1]^2$ in the unit square such that $\phi(0)=(0,0)$, $\phi(1)=(1,1)$, $\psi(0)=(0,1)$ and $\psi(1)=(1,0)$. By continuity of $\phi$ and $\psi$ there is a point of intersection. We prove that from…

Motivated by the fact that in a space where shortest paths are unique, no two shortest paths meet twice, we study a question posed by Greg Bodwin: Given a geodetic graph $G$, i.e., an unweighted graph in which the shortest path between any…

The minimum number of self-intersection points for members of a free homotopy class of curves on the punctured torus is bounded above in terms of the number L of letters required for a minimal description of the class in terms of the…

In this paper we study geometric coincidence problems in the spirit of the following problems by B. Gr\"unbaum: How many affine diameters of a convex body in $\mathbb R^n$ must have a common point? How many centers (in some sense) of…

Intersecting codes are a classical object in coding theory whose rank-metric analogue has recently been introduced. Although the definition formally parallels the Hamming-metric case, the structure and parameter constraints of rank-metric…

In this thesis we study in detail the self-intersection properties of Random Walks. Although notoriously hard to tackle, these properties are crucially related to the excluded-volume effect and other central features of real polymers. Our…

We define the crossing number for an embedding of a graph G into R^3, and prove a lower bound on it which almost implies the classical crossing lemma. We also give sharp bounds on the space crossing numbers of pseudo-random graphs.

In an orientable surface with boundary, free homotopy classes of curves on surfaces are in one to one correspondence with cyclic reduced words in a set of standard generators of the fundamental group. The combinatorial length of a class is…

We give a tight bound for the triple intersection numbers of Paley graphs. In particular, we show that any three vertices have a common neighbor in Paley graphs of order larger than 25.

We prove that on a restricted contact type hypersurface the number of leaf-wise intersections is bounded from below by a certain cup-length.

We introduce the concept of a pants decomposition for a finitely generated free group and construct the corresponding pants graph. A pants decomposition of a free group leads to the formation of a simplicial graph, referred to as the pants…

A self-avoiding polygon is a lattice polygon consisting of a closed self-avoiding walk on a square lattice. Surprisingly little is known rigorously about the enumeration of self-avoiding polygons, although there are numerous conjectures…