We characterize all of the ways to represent the wheel matroids and whirl matroids using frame matroids of signed graphs. The characterization of wheels is in terms of topological duality in the projective plane and the characterization of whirls is in terms of topological duality in the annulus.
& Qin, H.
(2008). The Signed-Graphic Representations of Wheels and Whirls. Discrete Mathematics, 308 (10), 1816-1825.