  We consider tree-dimensional domino tilings of cylinders $\mathcal{D} \times
[0,N] \subset \mathbb{R}^3$, where $\mathcal{D} \subset \mathbb{R}^2$ is a
balanced quadriculated disk and $N \in \mathbb{N}$. A flip is a local move in
the space of tilings: two adjacent and parallel dominoes are removed and then
replaced in a different position. The twist is a flip invariant that associates
an integer number to a domino tiling. A disk $\mathcal{D}$ is called regular if
any two tilings of $\mathcal{D} \times [0,N]$ sharing the same twist can be
connected trough a sequence of flips once extra vertical space is added to the
cylinder. We prove that hamiltonian disks with narrow and small bottlenecks are
regular. In particular, we show that the absence of a bottleneck in a
hamiltonian disk implies regularity.
