1,966 research outputs found
The Topology of Tile Invariants
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 of tiles
and a set of regions tileable by is isomorphic to a quotient of the second
homology group of a 2-complex built from . 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.Comment: 25 pages, 24 figure
Tilings of quadriculated annuli
Tilings of a quadriculated annulus A are counted according to volume (in the
formal variable q) and flux (in p). We consider algebraic properties of the
resulting generating function Phi_A(p,q). For q = -1, the non-zero roots in p
must be roots of unity and for q > 0, real negative.Comment: 33 pages, 12 figures; Minor changes were made to make some passages
cleare
Polyomino convolutions and tiling problems
We define a convolution operation on the set of polyominoes and use it to
obtain a criterion for a given polyomino not to tile the plane (rotations and
translations allowed). We apply the criterion to several families of
polyominoes, and show that the criterion detects some cases that are not
detectable by generalized coloring arguments.Comment: 8 pages, 8 figures. To appear in \emph{J. of Combin. Theory Ser. A
Tilings with T and Skew Tetrominoes
We consider tiling problems in the integer lattice. Specifically, we look at a set of four T-tetrominoes and four skew tetrominoes and determine when this set can tile rectangles and modified rectangles. Local considerations and coloring arguments are the main methods used to prove the untileability of regions
Principal minors and rhombus tilings
The algebraic relations between the principal minors of an matrix
are somewhat mysterious, see e.g. [lin-sturmfels]. We show, however, that by
adding in certain \emph{almost} principal minors, the relations are generated
by a single relation, the so-called hexahedron relation, which is a composition
of six cluster mutations.
We give in particular a Laurent-polynomial parameterization of the space of
matrices, whose parameters consist of certain principal and almost
principal minors. The parameters naturally live on vertices and faces of the
tiles in a rhombus tiling of a convex -gon. A matrix is associated to an
equivalence class of tilings, all related to each other by Yang-Baxter-like
transformations.
By specializing the initial data we can similarly parametrize the space of
Hermitian symmetric matrices over or the
quaternions. Moreover by further specialization we can parametrize the space of
\emph{positive definite} matrices over these rings
Ribbon Tilings and Multidimensional Height Functions
We fix and say a square in the two-dimensional grid indexed by
has color if . A {\it ribbon tile} of order is a
connected polyomino containing exactly one square of each color. We show that
the set of order- ribbon tilings of a simply connected region is in
one-to-one correspondence with a set of {\it height functions} from the
vertices of to satisfying certain difference restrictions.
It is also in one-to-one correspondence with the set of acyclic orientations of
a certain partially oriented graph.
Using these facts, we describe a linear (in the area of ) algorithm for
determining whether can be tiled with ribbon tiles of order and
producing such a tiling when one exists. We also resolve a conjecture of Pak by
showing that any pair of order- ribbon tilings of can be connected by a
sequence of local replacement moves. Some of our results are generalizations of
known results for order-2 ribbon tilings (a.k.a. domino tilings). We also
discuss applications of multidimensional height functions to a broader class of
polyomino tiling problems.Comment: 25 pages, 7 figures. This version has been slightly revised (new
references, a new illustration, and a few cosmetic changes). To appear in
Transactions of the American Mathematical Societ
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