Spaces of domino tilings |
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Authors: | N C Saldanha C Tomei M A Casarin Jr D Romualdo |
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Institution: | 1. Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina, 110 Jardim Botanico, 22460-320, Rio de Janeiro, RJ, Brazil 2. Departamento de Matemática, PUC-Rio, Rua Marquês de S?o Vicente, 225 Gávea, 22453-900, Rio de Janeiro, RJ, Brazil 3. Courant Institute of Mathematical Sciences, 251 Mercer Street, 10012, New York, NY, USA 4. Department of Economics, Harvard University, One Oxford Street, Science Center 325, 02138, Cambridge, MA, USA
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Abstract: | We consider the set of all tilings by dominoes (2×1 rectangles) of a surface, possibly with boundary, consisting of unit squares.
Convert this set into a graph by joining two tilings by an edge if they differ by aflip, i.e., a 90° rotation of a pair of side-by-side dominoes. We give a criterion to decide if two tilings are in the same connected
component, a simple formula for distances, and a method to construct geodesics in this graph. For simply connected surfaces,
the graph is connected. By naturally adjoining to this graph higher-dimensional cells, we obtain a CW-complex whose connected
components are homotopically equivalent to points or circles. As a consequence, for any region different from a torus or Klein
bottle, all geodesics with common endpoints are equivalent in the following sense. Build a graph whose vertices are these
geodesics, adjacent if they differ only by the order of two flips on disjoint squares: this graph is connected.
The first two authors received support from SCT and CNPq, Brazil. The other two were supported by a grant for undergraduates
by CNPq. |
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