arXiv subjects

Abdoul Karim Sane

Publications and source records attributed to Abdoul Karim Sane.

8 recordsLinked to original sources

arXiv2024

Rigidity on horocycles and hypercycles

We show that a bijection $f:\mathbb{H}^2\rightarrow\mathbb{H}^2$ of the hyperbolic plane that sends horocycles to horocycles (respectively hypercycles to hypercycles) is an isometry. This extends a previous result of J. Jeffers on geodesics to all curves with constant curvature in $\mathbb{H}^2$. We go beyond by showing that every abstract automorphism of the geodesic graph (respectively horocycles and hypercycles graphs) is induced by an earthquake map (respectively an isometry) of $\mathbb{H}^2$. This shadowed the difference between the geometry of geodesics and that of horocycles/hypercycles.

math.GT↗

arXiv2023

Connected components of the topological surgery graph of a unicellular collection

A unicellular collection on a surface is a collection of curves whose complement is a single disk. There is a natural surgery operation on unicellular collections, endowing the set of such with a graph structure where the edge relation is given by surgery. Here we determine the connected components of this graph, showing that they are enumerated by a certain homological "surgery invariant". Our approach is group-theoretic and proceeds by understanding the action of the mapping class group on unicellular collections. In the course of our arguments, we determine simple generating sets for the stabilizer in the mapping class group of a mod-$2$ homology class, which may be of independent interest.

math.GT↗

arXiv2022

Triangles, Fractales and Spaghetti

There is well-known problem of geometric probability which can be quote as the Broken Spaghetti Problem. It addresses the following question: A stick of spaghetti breaks into three parts and all points of the stick have the same probability to be a breaking point. What is the probability that the three sticks, putting together, form a triangle? In these notes, we describe a hidden geometric pattern behind the symmetric version of this problem, namely a fractal that parametrizes the sample space of this problem. Using that fractal, we address the question about the probability to obtain a $δ$-equilateral triangle.

math.HO↗

arXiv2021

Combinatorial $k$-systoles on a punctured torus and a pair of pants

In this paper, $S$ denotes a surface homeomorphic to a punctured torus or a pair of pants. Our interest is the study of \emph{\textbf{combinatorial $k$-systoles}} that is closed curves with self-intersection numbers greater than $k$ and with least combinatorial length. We show that the maximal intersection number $I^c_k$ of combinatorial $k$-systoles of $S$ grows like $k$ and $\underset{k\rightarrow+\infty}{\limsup}(I^c_k-k)=+\infty$. This result, in case of a pair of pants and a punctured torus, is a positive response to the combinatorial version of the Erlandsson - Parlier conjecture, originally formulated for the geometric length.

math.GT↗

arXiv2020

On dual unit balls of Thurston norms

Thurston norms are invariants of 3-manifolds defined on their second homology vector spaces, and understanding the shape of their dual unit ball is a (widely) open problem. W. Thurston showed that every symmetric polygon in Z^2, whose vertices satisfy a parity property, is the dual unit ball of a Thurston norm on a 3-manifold. However, it is not known if the parity property on the vertices of polytopes is a sufficient condition in higher dimension or if their are polytopes, with mod 2 congruent vertices, that cannot be realized as dual unit balls of Thurston norms. In this article, we provide a family of polytopes in Z^2g that can be realized as dual unit balls of Thurston norms on 3-manifolds. These polytopes come from intersection norms on oriented closed surfaces and this article widens the bridge between these two norms.

math.GT↗

arXiv2020

Unicellular maps and filtrations of the mapping class group

This article first answers to questions about connectedness of a new family of graphs on unicellular maps. Answering these questions goes through a description of the mapping class group as surgeries on unicellular maps. We also show how unicellular maps encode subgroups of the mapping group and provide filtrations of the mapping class group. These facts add a layer on the ubiquitous character of unicellular maps.

math.CO↗

arXiv2019

Curves on surfaces and surgeries

We study collections of curves in generic position on a closed surface whose complement consists of one disk only, up to orientation-preserving homeomorphism of the surface. We define a surgery operation on the set of such collections and prove that any two of them can be connected by a sequence of such surgeries.

math.GT↗

arXiv2018

Intersection Norms and One-faced Collection of Curves

Intersection norms are integer norms on the first homology group of a surface. In this article, we prove that there are some polytopes which are not dual unit balls of such norms. By the way, we investigate the set of collections of curves on $Σ$2 whose complement is a disk.

math.GT↗