Almost balanced biased graph representations of frame matroids

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