  We prove affirmatively the conjecture raised by J. Mostovoy; the space of
short ropes is weakly homotopy equivalent to the classifying space of the
topological monoid (or category) of long knots in $\mathbb{R}^3$. We make use
of techniques developed by S. Galatius and O. Randal-Williams to construct a
manifold space model of the classifying space of the space of long knots, and
we give an explicit map from the space of short ropes to the model in a
geometric way.
