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.