### Impossible configurations for geodesics on negatively-curved surfaces

Hass and Scott's example of a 4-valent graph on the 3-punctured sphere that cannot be realized by geodesics in any metric of negative curvature is generalized to impossible configurations filling surfaces of genus n with k punctures for any n and k.

Israel Journal of Mathematics volume 245, pages 165–172 (2021)
Slightly longer
arXiv version.

Here are some examples:

1. (a.) On the 3-punctured sphere. Vertices=3, Tracks=1.
This is the Hass-Scott
example.

(b.) On the punctured torus. Vertices=3, Tracks=3.

2. On the punctured torus. (a.) Vertices=7, Tracks=1.
(b.) Vertices=8, Tracks=2.

3. On the 2-holed torus. Vertices=9, Tracks=1.

*Anthony Phillips*

tony AT math.stonybrook.edu

September 28, 2021