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The same thing twice — page 14

Two constructions that look unrelated and turn out to be the same object wearing different clothes.

Every way to cut a genus-2 surface into pairs of pants. 2 thickened graphs, one for each type of pants decomposition of the closed surface of genus 2, each with 2 coloured junctions and 3 cutting circles.

Every surface is sewn from pants

A sphere with three holes — a pair of pants — is the smallest piece a surface with two or more handles can be cut into. Every such surface falls apart into them, and the Euler characteristic alone says how many pieces and how many cuts, however the cutting is done. What it does not say is the pattern, and the patterns are counted by drawing each one as a graph: two for two handles, five for three, seventeen for four.

The seven-vertex torus, unrolled onto a lattice. A triangular lattice with every point labelled a + 3b mod 7 and fourteen triangles shaded as one copy of the torus.

Topology

The fewest corners a surface needs

Build a closed surface from triangles, any two meeting along a whole edge, at a single corner or not at all, and ask for the fewest corners. Two lines of counting give a floor for every surface, in terms of its Euler characteristic alone. The torus meets it with seven, the projective plane with six — and the Klein bottle, which the counting says could be built from seven, cannot, as a search of every possible arrangement shows.

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