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

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Unformatted text preview: Computational Topology Homework 2 (due 10 / 27 / 09) Fall 2009 1. Consider a polygonal schema Π with a single face and n edges. Let Λ be the set of edge labels, and let ¯ Λ = { ¯ x | x ∈ Λ } . The signature of Π is a word in (Λ ∪ ¯ Λ) * 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 Σ(Π) . Let w 1 , w 2 ,..., w k be words in Λ * , such that each symbol in Λ 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. Euler’s 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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hw2 - Computational Topology Homework 2(due 10 27 09 Fall...

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