  The shrunk loop theorem presented here is an integral identity which
facilitates the calculation of the relative probability (or probability
amplitude) of any given topology that a free, closed Brownian or Feynman path
of a given 'duration' might have on the twice punctured plane (the plane with
two marked points). The result is expressed as a scattering series of integrals
of increasing dimensionality based on the maximally shrunk version of the path.
Physically, this applies in different contexts: (i) the topology probability of
a closed ideal polymer chain on a plane with two impassable points, (ii) the
trace of the Schroedinger Green function, and thence spectral information, in
the presence of two Aharonov-Bohm fluxes, (iii) the same with two branch points
of a Riemann surface instead of fluxes. Our theorem starts with the Stovicek
expansion for the Green function in the presence of two Aharonov-Bohm flux
lines, which itself is based on the famous Sommerfeld one puncture point
solution of 1896 (the one puncture case has much easier topology, just one
winding number). Stovicek's expansion itself can supply the results at the
expense of choosing a base point on the loop and then integrating it away. The
shrunk loop theorem eliminates this extra two dimensional integration,
distilling the topology from the geometry.
