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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 2manifold () . 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 2manifold has a polygonal schema with signature w 1 w R 1 w 2 w R 2 w k w R k ? (b) Which 2manifold 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?...
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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.
 Fall '08
 Staff

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