  We consider manifolds with conic singularites that are isometric to
$\mathbb{R}^{n}$ outside a compact set. Under natural geometric assumptions on
the cone points, we prove the existence of a logarithmic resonance-free region
for the cut-off resolvent. The estimate also applies to the exterior domains of
non-trapping polygons via a doubling process. The proof of the resolvent
estimate relies on the propagation of singularities theorems of Melrose and the
second author to establish a "very weak" Huygens' principle, which may be of
independent interest. As applications of the estimate, we obtain a exponential
local energy decay and a resonance wave expansion in odd dimensions, as well as
a lossless local smoothing estimate for the Schr{\"o}dinger equation.
