  We study translating solitons for the mean curvature flow,
$\Sigma^2\subseteq\mathbb{R}^3$ which are contained in slabs, and are of finite
genus and finite entropy. As a first consequence of our results, we can
enumerate connected components of slices to define asymptotic invariants
$\omega^\pm(\Sigma)\in\mathbb{N}$, which count the numbers of "wings''.
Analyzing these, we give a method for computing the entropies $\lambda(\Sigma)$
via a simple formula involving the wing numbers, which in particular shows that
for this class of solitons the entropy is quantized into integer steps.
Finally, combining the concept of wing numbers with Morse theory for minimal
surfaces, we prove the uniqueness theorem that if $\Sigma$ is a complete
embedded simply connected translating soliton contained in a slab with entropy
$\lambda(\Sigma)=3$ and containing a vertical line, then $\Sigma$ is one of the
translating pitchforks of Hoffman-Mart\'in-White
