CS 39R: Symmetry & Topology
Lecture #6 -- Mon. 3/07, 2016.

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Preparation:

Construct a physical model of a surface of genus 2,
and then draw onto this surface TWO closed loops that together still leave the surface fully connected (just ONE domain).

Warm-up:

Some TRUE/FALSE test questions:

T/F?: The number of punctures in a surface does not affect its genus... ?

T/F?: Single sides surfaces cannot have borders... ?

T/F?: The genus of a single-sided surface is typically higher than that of a 2-sided surface... ?

First, try to answer these questions by yourself, -- then compare your findings with your neighbors.

The Topology of Single-sided, Non-orientable 2-Manifolds (cont.)

By now you should be familiar with the concept of Moebius bands and Klein bottles of various shapes.
At the end of last lecture you learned about the Surface Classification Theorem relying on just three parameters.
Today we will deepen our understanding of these concepts and learn a method to construct any type of 2-manifold:

More extensive PPT talk: "Cross-Caps -- Boy Caps -- Boy Cups"
(Some of this will be a repetition of things I have shown you before, but hopefully this helps to tie the various concepts together.)

Surfaces with Holes and Boundaries -- Key Concepts:

If we allow surfaces to have "punctures" or "holes" -- which then have "boundaries" or "rims"
-- things get a little more complicated.
But a topologist can still classify all the possible surfaces of that kind by only three characteristics:

ORIENTABILITY: Is the surface two-side (orientable) or single-sided (non-orientable)?

# OF BOUNDARY COMPONENTS: How many "disks" have been removed from a closed surface;
or, how many individual rims or hole contours, h, are there?

EULER CHARACTERISTIC, X (or alternatively, its GENUS, g): How "connected" is the surface?
X = #Vertices - #Edges + #Facets of a mesh approximating the surface.

What are surfaces of higher genus good for?

-- among other things: to embed complex graphs without any crossings.

Elements of a graph (G):
a set of Vertices: V(G) = {U, V, W, X, Y, Z ...}. [these are their names].
a set of Edges: E(G) = {(U,V), (V,W), (X,Y), (X,Z) ... }. [defined by their end points].
-- Loops are OK, e.g.: (U,U).
-- Multiple edges between vertices are OK, e.g.: {(V,W), (V,W), ...).

A complete graph, K#, has exactly one edge between every possible pair of vertices.
K2 has 1 edge; K3 has 3 edges; K4 has 6 edges; K5 has 10 edges; K6 has 15 edges; K7 has 21 edges ...

A planar graph is one that can be drawn crossing-free into the plane.

New Homework Assignments:

Due: March 14, 2016
1.) Determine which of the complete graphs listed above are planar.
For the non-planar graphs find out what surface of lowest possible genus allows them to be drawn crossing-free.
Bring sketches or models of your solution to class.

2.) Refine your course project proposals:
State some explicit goals for what you are looking for and how you plan to analyze and use these findings.
Send your refined proposals to me by e-mail before noon on Monday, March 14, 2013.