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Séminaire Lotharingien de Combinatoire, 82B.39 (2019), 12 pp.

# Spencer Backman, Francisco Santos, and Chi Ho Yuen

# Topological bijections for oriented matroids

**Abstract.**
In previous work by the first and third author with Matthew Baker, a family of bijections between bases of a regular matroid and the Jacobian group of the matroid was given. The core of the work is a geometric construction using zonotopal tilings that produces bijections between the bases of a realizable oriented matroid and the set of (σ,σ^{*})-*compatible orientations* with respect to some *acyclic* circuit (respectively, cocircuit) signature σ (respectively, σ^{*}). 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 *T*_{M}(1,1) using orientation activity due to Gioan and Las Vergnas. Here *T*_{M}(*x*,*y*) is the Tutte polynomial of the matroid.

Received: November 15, 2018.
Accepted: February 17, 2019.
Final version: April 1, 2019.

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