Exceptional Balanced Triangulations on Surfaces

Izmestiev, Klee and Novik proved that any two balanced triangulations of a closed surface \(F^2\) can be transformed into each other by a sequence of six operations called basic cross flips. Recently Murai and Suzuki proved that among these six operations only two operations are almost sufficient in the sense that, with for finitely many exceptions, any two balanced triangulations of a closed surface \(F^2\) can be transformed into each other by these two operations. We investigate such finitely many exceptions, called exceptional balanced triangulations, and obtain the list of exceptional balanced triangulations of closed surfaces with low genera. Furthermore, we discuss the subsets \(\mathcal{O}\) of the six operations satisfying the property that any two balanced triangulations of the same closed surface can be connected through a sequence of operations from \(\mathcal{O}\).