  This paper constructs coherent states for spin networks with planar symmetry.
After gauge-fixing, the full SU(2) symmetry is broken to U(1), but one cannot
simply use the U(1) limit of SU(2) coherent states, because the planar states
exhibit an unexpected O(3) symmetry arising from the closed loop character of
the transverse directions. The coherent states constructed in this paper obey
this symmetry. They are superpositions of holonomies which obey the residual
U(1) symmetry only on average; some holonomies in the superposition violate the
symmetry, although the U(1) quantum numbers of these holonomies are peaked at
values which obey the symmetry. Operators acting on coherent states give back a
c-number times the original state, plus small correction states, which make the
coherent state an approximate, rather than exact eigenstate of the operator. In
a follow-on paper, these small correction states are used to calculate small
corrections to the volume operator.
