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The shrunk loop theorem presented here is an integral identity whichfacilitates the calculation of the relative probability (or probabilityamplitude) of any given topology that a free, closed Brownian or Feynman pathof a given 'duration' might have on the twice punctured plane (the plane withtwo marked points). The result is expressed as a scattering series of integralsof increasing dimensionality based on the maximally shrunk version of the path.Physically, this applies in different contexts: (i) the topology probability ofa closed ideal polymer chain on a plane with two impassable points, (ii) thetrace of the Schroedinger Green function, and thence spectral information, inthe presence of two Aharonov-Bohm fluxes, (iii) the same with two branch pointsof a Riemann surface instead of fluxes. Our theorem starts with the Stovicekexpansion for the Green function in the presence of two Aharonov-Bohm fluxlines, which itself is based on the famous Sommerfeld one puncture pointsolution of 1896 (the one puncture case has much easier topology, just onewinding number). Stovicek's expansion itself can supply the results at theexpense of choosing a base point on the loop and then integrating it away. Theshrunk loop theorem eliminates this extra two dimensional integration,distilling the topology from the geometry.Comment: 29 pages, 5 figures (accepted by J. Phys. A: Math. Gen.

Shrunk loop theorem for the topology probabilities of closed Brownian (or Feynman) paths on the twice punctured plane