  In the presence of certain topological conditions, we provide lower bounds
for the infimum of the length function associated to a collection of curves on
Teichm\"uller space that depend on the dual cube complex associated to the
collection, a concept due to Sageev. As an application of our bounds, we obtain
estimates for the `longest' curve with $k$ self-intersections, complementing
work of Basmajian \cite{basmajian}.
