  We prove two theorems about homotopies of curves on 2-dimensional Riemannian
manifolds. We show that, for any epsilon > 0, if two simple closed curves are
homotopic through curves of bounded length L, then they are also isotopic
through curves of length bounded by L + epsilon. If the manifold is orientable,
then for any epsilon > 0 we show that, if we can contract a curve gamma
traversed twice through curves of length bounded by L, then we can also
contract gamma through curves bounded in length by L + epsilon.
  Our method involves cutting curves at their self-intersection points and
reconnecting them in a prescribed way. We consider the space of all curves
obtained in this way from the original homotopy, and use a novel approach to
show that this space contains a path which yields the desired homotopy.
