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Twists for duplex regions

This note relies heavily on arXiv:1404.6509 and arXiv:1410.7693. Both articles discuss domino tilings of three-dimensional regions, and both are concerned with flips, the local move performed by removing two parallel dominoes and placing them back in the only other possible position. In the second article, an integer $\operatorname{Tw}(t)$ is defined for any tiling $t$ of a large class of regions $\mathcal{R}$: it turns out that $\operatorname{Tw}(t)$ is invariant by flips. In the first article, a more complicated polynomial invariant $P_t(q)$ is introduced for tilings of two-story regions. It turns out that $\operatorname{Tw}(t) = P_t'(1)$ whenever $t$ is a tiling of a duplex region, a special kind of two-story region for which both invariants are defined. This identity is proved in arXiv:1410.7693 in an indirect and nonconstructive manner. In the present note, we provide an alternative, more direct proof.