8 research outputs found
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
Cut-and-paste of quadriculated disks and arithmetic properties of the adjacency matrix
We define cut-and-paste, a construction which, given a quadriculated disk
obtains a disjoint union of quadriculated disks of smaller total area. We
provide two examples of the use of this procedure as a recursive step. Tilings
of a disk receive a parity: we construct a perfect or near-perfect
matching of tilings of opposite parities. Let be the black-to-white
adjacency matrix: we factor , where and are
lower and upper triangular matrices, is obtained from a larger
identity matrix by removing rows and columns and all entries of ,
and are equal to 0, 1 or -1.Comment: 20 pages, 17 figure
Domino tilings of three-dimensional regions: flips, trits and twists
In this paper, we consider domino tilings of regions of the form , where is a simply connected planar region and . It turns out that, in nontrivial examples, the set of such
tilings is not connected by flips, i.e., the local move performed by removing
two adjacent dominoes and placing them back in another position. We define an
algebraic invariant, the twist, which partially characterizes the connected
components by flips of the space of tilings of such a region. Another local
move, the trit, consists of removing three adjacent dominoes, no two of them
parallel, and placing them back in the only other possible position: performing
a trit alters the twist by . We give a simple combinatorial formula for
the twist, as well as an interpretation via knot theory. We prove several
results about the twist, such as the fact that it is an integer and that it has
additive properties for suitable decompositions of a region.Comment: 38 pages, 17 figures. Most of this material is also covered in the
first author's Ph.D. thesis (arXiv:1503.04617