  We present two models for the space of knots which have endpoints at fixed
boundary points in a manifold with boundary, one model defined as an inverse
limit of spaces of maps between configuration spaces and another which is
cosimplicial. These models build on the calculus of isotopy functors and are
weakly homotopy equivalent to knot spaces when the ambient dimension is greater
than three. The mapping space model, and the evaluation map on which it builds,
is suitable for analysis through differential topology. The cosimplicial model
gives rise to spectral sequences which converge to cohomology and homotopy
groups of spaces of knots when they are connected. We explicitly identify and
establish vanishing lines in these spectral sequences.
