Almost balanced biased graph representations of frame matroids

Given a 3-connected biased graph $\Omega$ with a balancing vertex, and with frame matroid $F(\Omega)$ nongraphic and 3-connected, we determine all biased graphs $\Omega'$ with $F(\Omega') = F(\Omega)$. As a consequence, we show that if $M$ is a 4-connected nongraphic frame matroid represented by a biased graph $\Omega$ having a balancing vertex, then $\Omega$ essentially uniquely represents $M$. More precisely, all biased graphs representing $M$ are obtained from $\Omega$ by replacing a subset of the edges incident to its unique balancing vertex with unbalanced loops.