In this paper we consider domino tilings of bounded regions in dimension n≥4. We define the twist of such a tiling, an elements of
Z/(2), and prove it is invariant under flips, a simple local move
in the space of tilings.
We investigate which regions D are regular, i.e. whenever two tilings t0
and t1 of D×[0,N] have the same twist then t0 and t1 can be
joined by a sequence of flips provided some extra vertical space is allowed. We
prove that all boxes are regular except D=[0,2]3.
Furthermore, given a regular region D, we show that there exists a value
M (depending only on D) such that if t0 and t1 are tilings of equal
twist of D×[0,N] then the corresponding tilings can be joined by a
finite sequence of flips in D×[0,N+M]. As a corollary we deduce that,
for regular D and large N, the set of tilings of D×[0,N] has two
twin giant components under flips, one for each value of the twist.Comment: 28 pages, 14 figure