[Math] How to compute Euler characteristic from polygonal presentation

algebraic-topology

How can I compute the Euler characteristic of a compact surface from its polygonal presentation $\langle S | W_1 , \ldots , W_k \rangle$? I guess that the number of edges is the number of different symbols in $S$, but how to get the number of vertices and the number of faces?

Best Answer

If the polygonal presentation contains only one word, the following algorithm works:

  1. Add vertex symbols. Then, for each pair of edges with the same edge symbol, build one set containing the two initial vertices and one set containing the two terminal vertices.
  2. Build the union of all vertex sets that have at least one vertex in common until all resulting sets are disjoint.
  3. The number of vertices is the number of disjoint sets obtained in step 2. The number of edges is the number of different edge symbols and the number of faces is one.

Example:

Cube: $S = \langle a, b, c, d, e, f \mid aa^{-1}e^{-1}dd^{-1}gcc^{-1}g^{-1}e f bb^{-1}f^{-1} \rangle$ (c.f. A.P.'s answer)

Adding vertex symbols: $AaBa^{-1}Ce^{-1}DdEd^{-1}FgGcHc^{-1}Ig^{-1}KeLfM bNb^{-1}Of^{-1}A$

Building vertex sets:

Edge $a$: $\lbrace A, C \rbrace$, $\lbrace B, B \rbrace$; Edge $b$: $\lbrace M, O \rbrace$, $\lbrace N, N \rbrace$; Edge $c$: $\lbrace G, I \rbrace$, $\lbrace H, H \rbrace$; Edge $d$: $\lbrace D, F \rbrace$, $\lbrace E, E \rbrace$; Edge $e$: $\lbrace D, K \rbrace$, $\lbrace C, L \rbrace$; Edge $f$: $\lbrace L, A \rbrace$, $\lbrace M, O \rbrace$; Edge $g$: $\lbrace F, K \rbrace$, $\lbrace G, I \rbrace$

Building unions:

$\lbrace A, C \rbrace \cup \lbrace C, L \rbrace \cup \lbrace L, A \rbrace = \lbrace A, C, L \rbrace$, $\lbrace D, F \rbrace \cup \lbrace D, K \rbrace \cup \lbrace F, K \rbrace = \lbrace D, F, K \rbrace$

Resulting disjoint vertex sets:

$\lbrace A, C, L \rbrace$, $\lbrace D, F, K \rbrace$, $\lbrace M, O \rbrace$, $\lbrace G, I \rbrace$, $\lbrace B \rbrace$, $\lbrace E \rbrace$, $\lbrace H \rbrace$, $\lbrace N \rbrace$

The number of disjoint vertex sets is 8. Thus, the number of vertices is 8.

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