Closed curves and geodesics with two self-intersections on the punctured torus
type: Item
instance of: scholarly article
title: Closed curves and geodesics with two self-intersections on the punctured torus
zbMATH Open document ID: 0901.57022
DOI: 10.1007/BF01317313
author: Susan Dziadosz
author: Dennis J. Garity
author: Thomas Insel
author: Peter Wiles
author: David James Crisp
author: Thomas A. Schmidt
published in: Monatshefte für Mathematik
publication date: 1998-07-15T00:00:00Z
full work available at URL: https://eudml.org/doc/178807
review text: This paper is an outgrowth of an NSF REU project at Oregon State University. In Sections 1 and 2, the topological, geometric and number theoretic background for this project is given. In Section 3, the classification up to homeomorphism of the free homotopy classes of closed curves on a once-punctured torus \(T\), whose self-intersection number is two, is obtained. There are eight such classes, six primitive and two imprimitive. The case analysis involved uses standard combinatorial topological methods. The proof that the classes obtained are distinct is done in Section 6. The topological and geometric consequences of the classification are given in Sections 4 and 5. Thus, up to homeomorphism, there is a unique curve in each class realizing the minimum self-intersection number. Given a hyperbolic metric on \(T\), the classification of geodesics on \(T\) with two self-intersections is obtained. Some new results on the Markov spectrum of diophantine approximation are derived in Section 7. Finally, some questions and suggestions for further research are pointed out.
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 11J06
Mathematics Subject Classification ID: 30F99
zbMATH DE Number: 1154520
zbMATH Keywords: geodesic
zbMATH Keywords: punctured torus
zbMATH Keywords: Markov spectrum
zbMATH Keywords: diophantine approximation
reviewed by: Mircea Craioveanu
MaRDI profile type: Publication
cites work: Geometry and spectra of compact Riemann surfaces
cites work: The geometry and spectrum of the one holed torus
cites work: Q3140889
cites work: Q3140890
cites work: Q3993084
cites work: Diophantine approximation on hyperbolic orbifolds
cites work: Parameters for Fuchsian groups II: topological type (1,1)
cites work: Q4877336
cites work: Minimum of quadratic forms with respect to Fuchsian groups. I.
cites work: The geometry of Markoff numbers
cites work: Characterization of simple closed geodesics on Fricke surfaces
cites work: On equivalent sets of elements in a free group
cites work: On the Kähler form of the moduli space of once punctured tori
OpenAlex ID: W2073316766
Recommended article: Self-Intersection Numbers of Curves on the Punctured Torus
Recommended article: Short closed geodesics with self-intersections
Recommended article: On the number of closed geodesics on the 2-torus
Recommended article: Q3210488
Recommended article: Angular self-intersections for closed geodesics on surfaces
Recommended article: ON THE EXISTENCE OF CLOSED GEODESICS ON TWO-SPHERES
Recommended article: Combinatorics of simple closed curves on the twice punctured torus
Recommended article: Characterization and computation of closed geodesics on toroidal surfaces
Recommended article: On pairs of closed geodesics on hyperbolic surfaces
Recommended article: Closed geodesics on incomplete surfaces
description: scientific article; zbMATH DE number 1154520