Theorem \[theorem:twist number = complexity\] the following corollary follows immediately.

\[coro:well definedness of twist number\] The twist number of a Bott manifold $M$ is well-defined, i.e., any two Bott tower structures of $M$ have the same twist number.

BQ-algebras and Bott manifolds
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Recall that a $2n$-dimensional manifold $M$ is a quasitoric manifold over a simple (combinatorial) polytope $P$ if there is a locally standard $n$-torus $T^n$ action on $M$ and a surjective map $\pi:M\to P$ whose fibers are the $T^n$-orbits. For a $2n$-dimensional quasitoric manifold $M$ over a simple polyotpe $P$ there corresponds a [*characteristic map*]{} $\chi:\mathcal F \to \mathbb Z^n$ well-defined up to sign where $\mathcal F$ is the set of all facets of $P$. A characteristic map should satisfy the following two conditions:

-   $\chi(F)$ is a primitive vector for any $F\in\mathcal F$, and

-   if $n$ facets $F_1\ldots, F_n$ are intersecting at vertex $v$ of $P$, then $\{\chi(F_1),\ldots, \chi(F_n)\}$ forms a linearly independent subset in $\mathbb Z^n$.

Conversely, for simple polytope $P$ and a map $\chi:\mathcal F \to \mathbb Z^n$ satisfying the above two conditions, there exists a unique quasitoric manifold up equivalence whose characteristic map is $\chi$.

Two quasitoric manifolds $\pi_M: M\to P$ and $\pi_N:N\to 