  Equilibrium states of a closed semiflexible polymer binding to a cylinder are
described. This may be either by confinement or by constriction. Closed
completely bound states are labeled by two integers: the number of
oscillations, $n$, and the number of times it winds the cylinder, $p$, the
latter a topological invariant. We examine the behavior of these states as the
length of the loop is increased by evaluating the energy, the conserved axial
torque and the contact force. The ground state for a given $p$ is the state
with $n=1$; a short loop with $p=1$ is an elliptic deformation of a parallel;
as its length increases it elongates along the cylinder axis, with two hairpin
ends. Excited states with $n\geq2$ and $p=1$ possess $n$-fold axial symmetry.
Short (long) loops possess energies $\approx p E_0$ ($nE_0$), with $E_0$ the
energy of a circular loop with same radius as the cylinder; in long loops the
axial torque vanishes. Confined bound excited states are initially unstable;
however, above a critical length each $n$-fold state becomes stable: the folded
hairpin cannot be unfolded. The ground state for each $p$ is also initially
unstable with respect to deformations rotating the loop off the surface into
the interior. A closed planar elastic curve aligned along the cylinder axis
making contact with the cylinder on its two sides is identified as the ground
state of a confined loop. Exterior bound states behave very differently, if
free to unbind, as signaled by the reversal in the sign of the contact force.
If $p=1$, all such states are unstable. If $p \geq 2$, however, a topological
obstruction to complete unbinding exists. If the loop is short, the bound state
with $p=2$ and $n=1$ provides a stable constriction of the cylinder, partially
unbinding as the length is increased. This motif could be relevant to an
understanding of the process of membrane fission mediated by dynamin rings.
