Notes

Notes

Notes will be posted occasionally to supplement the material in lectures and references.

Surfaces and Turns

Tuesday, Jan 28, 2020.

In this note we describe how to determine if a two-complex is an oriented surface (possibly with boundary), with polygons oriented according to the orientation of the surface. Some cases may be missed below, so please check the description in addition to implementing it. Assume that we are given a two-complex complex: TwoComplex. First steps Check that the complex is valid, using complex.checkComplex (also check if that method covers all checks).

Quadrangulations: Constructions and Geodesics

Wednesday, Jan 29, 2020.

We previously worked with general two-complexes, including those representing surfaces. We now specialize to the first of the three special representations that are most useful to us - non-positive quadrangulations. Here we discuss what these are, how to construct these and the geodesics for these. Non-positive quadrangulations For a closed surface $\Sigma$, a non-positive quadrangulation is a two-complex homeomorphic to $\Sigma$ such that Every face has four sides. The degree of (i.

Intersections of Combinatorial Geodesics (following Despre-Lazarus)

Thursday, Mar 19, 2020.

This is a sketch of the description of intersection numbers in the reference paper, including why it works. I have not checked some of the technical lemmas. I also focus on the basic case - different, primitive curves in a closed, oriented surface. I will mainly just describe stuff. Geometric intersections and geodesics The guiding example is the description of intersection numbers in terms of hyperbolic geometry. The classic reference is the book Automorphisms of Surfaces after Nielsen and Thurston by Casson and Bleiler.