Related papers: Self intersections on a pair of pants

We construct immersions of trivalent abstract tropical curves in the Euclidean plane and embeddings of all abstract tropical curves in higher dimensional Euclidean space. Since not all curves have an embedding in the plane, we define the…

We investigate the intersection body of a convex polytope using tools from combinatorics and real algebraic geometry. In particular, we show that the intersection body of a polytope is always a semialgebraic set and provide an algorithm for…

We prove a bound for the geodesic diameter of a subset of the unit ball in $\mathbb{R}^n$ described by a fixed number of quadratic equations and inequalities, which is polynomial in $n$, whereas the known bound for general degree is…

A geometric intersection graph is constructed over a set of geometric objects, where each vertex represents a distinct object and an edge connects two vertices if and only if the corresponding objects intersect. We examine the problem of…

Let $S_{g,p}$ denote the genus $g$ orientable surface with $p \ge 0$ punctures, and let $\omega(g,p)= 3g+p-4$. We prove the existence of infinitely long geodesic rays $\left\{v_{0},v_{1}, v_{2}, ...\right\}$ in the curve graph satisfying…

According to a conjecture of Pach, there are $O(n)$ tangent pairs among any family of $n$ Jordan arcs in which every pair of arcs has precisely one common point and no three arcs share a common point. This conjecture was proved for two…

We prove that, on any closed manifold of dimension at least two with non-trivial first Betti number, a $C^\infty$ generic Riemannian metric has infinitely many closed geodesics, and indeed closed geodesics of arbitrarily large length. We…

We introduce the \textit{modular intersection kernel}, and we use it to study how geodesics intersect on the full modular surface $\mathbb{X}=PSL_2\left(\mathbb{Z}\right) \backslash \mathbb{H}$. Let $C_d$ be the union of closed geodesics…

Given two distinct reduced, irreducible curves of given degrees, contained in projective space but whose union is not contained in a hyperplane, what is the largest number of points of intersection they can have? When the projective space…

We give an explicitly computable lower bound for the arithmetic self-intersection number of the dualizing sheaf on a large class of arithmetic surfaces. If some technical conditions are satisfied, then this lower bound is positive. In…

We prove an upper bound for the number of shortest closed geodesics in a closed hyperbolic manifold of any dimension in terms of its volume and systole, generalizing a theorem of Parlier for surfaces. We also obtain bounds on the number of…

John E. Littlewood posted the question {\em ``Is it possible in 3-space for seven infinite circular cylinders of unit radius each to touch all the others? Seven is the number suggested by counting constants.''} Boz\'oki, Lee, and R\'onyai…

Given a hyperbolic surface $\S$, a classic result of Birman and Series states that for each $K$, all complete geodesics with at most $K$ self-intersections can only pass through a certain nowhere dense, Hausdorff dimension 1 subset of $\S$.…

We show that all geodesic rays in the uniform infinite half-planar quadrangulation (UIHPQ) intersect the boundary infinitely many times, answering thereby a recent question of Curien. However, the possible intersection points are sparsely…

We introduce a two-dimensional lattice model for the description of knotted polymer rings. A polymer configuration is modeled by a closed polygon drawn on the square diagonal lattice, with possible crossings describing pairs of strands of…

We examine an equivalence relation between free homotopy classes of closed curves on the pair of pants known as k-equivalence, a generalization of a concept previously defined by Leininger. We prove that two classes of closed curves on the…

We introduce the cycle intersection graph of a graph, an adaptation of the cycle graph of a graph, and use the structure of these graphs to prove an upper bound for the decycling number of all even graphs. This bound is shown to be…

We modify an approach of Johnson to define the distance of a bridge splitting of a knot in a 3-manifold using the dual curve complex and pants complex of the bridge surface. This distance can be used to determine a complexity, which becomes…

Three intersection theorems are proved. First, we determine the size of the largest set system, where the system of the pairwise unions is l-intersecting. Then we investigate set systems where the union of any s sets intersect the union of…

Gao and Richter (1994) showed that every $3$-connected graph which embeds on the plane or the projective plane has a spanning closed walk meeting each vertex at most $2$ times. Brunet, Ellingham, Gao, Metzlar, and Richter (1995) extended…