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arXiv · 1507.02315

The Topology of Tile Invariants

Michael P. Hitchman
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Abstract

In this note we use techniques in the topology of 2-complexes to recast some tools that have arisen in the study of planar tiling questions. With spherical pictures we show that the tile counting group associated to a set $T$ of tiles and a set of regions tileable by $T$ is isomorphic to a quotient of the second homology group of a 2-complex built from $T$. In this topological setting we derive some well-known tile invariants, one of which we apply to the solution of a tiling question involving modified rectangles.

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Michael P. Hitchman. 2015-07-08. The Topology of Tile Invariants. https://doi.org/10.1216/rmj-2015-45-2-539

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