  The energy minimization problem associated to uniform, isotropic, linearly
elastic rods leads to a geometric variational problem for the rod centerline,
whose solutions include closed, knotted curves. We give a complete description
of the space of closed and quasiperiodic solutions. The quasiperiodic curves
are parametrized by a two-dimensional disc. The closed curves arise as a
countable collection of one-parameter families, connecting the m-fold covered
circle to the n-fold covered circle for any m,n relatively prime. Each family
contains exactly one self-intersecting curve, one elastic curve, and one closed
curve of constant torsion. Two torus knot types are represented in each family,
and all torus knots are represented by elastic rod centerlines.
