Abstract: In previous work by the ?rst and third author with Matthew Baker, a family of bijections between bases of a regular matroid and the Jacobian group of the matroid wasgiven. Thecoreoftheworkisageometricconstructionusingzonotopaltilingsthat produces bijections between the bases of a realizable oriented matroid and the set of (?,??)-compatibleorientationswithrespecttosomeacycliccircuit(respectively, cocircuit) signature ? (respectively, ??). In this work, we extend this construction to general oriented matroids and circuit (respectively, cocircuit) signatures coming from generic single-element liftings (respectively, extensions). As a corollary, when both signatures are induced by the same lexicographic data, we give a new (bijective) proof of the interpretation of TM(1,1) using orientation activity due to Gioan and Las Vergnas. Here TM(x,y) is the Tutte polynomial of the matroid.

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