Homology of polyomino tilings on flat surfaces

Edin Lidjan,Ðordje Baralic

doi:10.2298/aadm210307031l

Abstract

The homology group of a tiling introduced by M. Reid is studied for certain topological tilings. As in the planar case, for finite square grids on topological surfaces, the method of homology groups, namely the non-triviality of some specific element in the group allows a "coloring proof" of impossibility of a tiling. Several results about the non-existence of polyomino tilings on certain square-tiled surfaces are proved in the paper.

Highlights

Recreational mathematics comprises various subjects including combinatorial games, puzzles, card tricks, art, etc
We study the problem of tiling a surface S subdivided into a finite ‘combinatorial’ grid by a finite set of polyomino shapes T and define the homology group HS(T )
We present some new results and illustrate examples explaining the application of the homology group of generalized polyomino type tilings in the combinatorial and the topological context

Summary

INTRODUCTION

Recreational mathematics comprises various subjects including combinatorial games, puzzles, card tricks, art, etc. Conway and Lagarias developed in [5] the so-called ‘boundary-word method’ for addressing this question Their ideas were further developed by Reid in [17] who assigned to each set of tiles T the homology and the homotopy group of tilings and formulated a necessary condition for existence of a proper tiling of a finite region M in a plane. We study the problem of tiling a surface S subdivided into a finite ‘combinatorial’ grid by a finite set of polyomino shapes T and define the homology group HS(T ). Square-tiled and translation surfaces arise in dynamical systems, where they can be used to model billiards, and in Teichmuller theory They have a rich mathematical structure and may be studied from multiple points of view (flat geometry, algebraic geometry, combinatorial group theory, etc.). Our main novelty lies in Theorem 12 which establishes a general result connecting the I-polyomino shape with the genus of the surface

TILING PROBLEM ON SURFACES

NONEXISTENCE OF POLYOMINO TILINGS ON SURFACES