Self-crossing geodesics

A rubber band is stretched around a pebble, and it crosses itself at several points (see Figure 1). The combinatorics of self-crossings can be described by a closed plane curve—it is the rubber band in a parameterization of the surface with one point removed. For example, if you could turn the pebble around, you would see that the self-crossings are described by the plane curve in Figure 1(b). We assume that the surface of the pebble is strongly convex, smooth, and frictionless; in this case, the rubber band models a closed geodesic. Suppose that we are interested in possible patterns of self-crossings. More precisely, what are the possible combinatoric types of self-crossings of a closed geodesic on a strongly convex smooth closed surface?

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Work Title Self-crossing geodesics
Open Access
  1. Anton Petrunin
License In Copyright (Rights Reserved)
Work Type Article
  1. Springer Science and Business Media LLC
Publication Date October 28, 2021
Publisher Identifier (DOI)
  1. 10.1007/s00283-021-10127-0
  1. The Mathematical Intelligencer
Deposited August 24, 2022




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