We prove affirmatively the conjecture raised by J Mostovoy (Topology 41 (2002) 435–450); the space of short ropes is weakly homotopy equivalent to the classifying space of the topological monoid (or category) of long knots in . We make use of techniques developed by S Galatius and O Randal-Williams (Geom. Topol. 14 (2010) 1243–1302) 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.
"The space of short ropes and the classifying space of the space of long knots." Algebr. Geom. Topol. 18 (5) 2859 - 2873, 2018. https://doi.org/10.2140/agt.2018.18.2859