hw2 - Computational Topology Homework 2 (due 10 / 27 / 09)...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Computational Topology Homework 2 (due 10 / 27 / 09) Fall 2009 1. Consider a polygonal schema with a single face and n edges. Let be the set of edge labels, and let = { x | x } . The signature of is a word in ( ) * describing the sequence of edges on its single face; each edge label appears exactly twice (possibly barred). The signature completely determines the homeomorphism type of the 2-manifold () . Let w 1 , w 2 ,..., w k be words in * , such that each symbol in appears exactly once in exactly one w i . Let w R denote the reversal of any word w , and let w denote the inverse of w , obtained by barring each letter in w R . Thus, if w = abc , then w R = cba , w = cba , and w R = abc . (a) Which 2-manifold has a polygonal schema with signature w 1 w R 1 w 2 w R 2 w k w R k ? (b) Which 2-manifold has a polygonal schema with signature w 1 w R 1 w 2 w R 2 w k w R k ? Prove your answers are correct. For example, abccbadeed is an example for part (a), and cbacba deed is an example for part (b), where w 1 = abc and w 2 = de . 2. Eulers formula relates the number of vertices, edges, and faces in a combinatorial surface to its Euler genus: V- E + F = 2- g . (Recall that g = 2 g if the surface is orientable and g = g otherwise.) (a) A triangulation is an embedded graph in which every facial walk has length 3. (For example, this is the simplest triangulation of the sphere: .) Let T be a triangulation with n vertices of a surface of Euler genus g . Exactly how many edges and triangles does T have?...
View Full Document

This note was uploaded on 01/24/2012 for the course CS 598 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.

Page1 / 3

hw2 - Computational Topology Homework 2 (due 10 / 27 / 09)...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online