  The experimental features of positive-parity doublet bands in the
odd-\emph{A} cesium isotopes $^{125,127,129,131}$Cs, including angular momentum
alignment, energy staggering, $B(M1)/B(E2)$ etc. are studied systematically and
compared to those of the candidate chiral bands in the adjacent odd-odd Cs
isotopes. The configuration assignments and the dynamics of these bands are
discussed. The self-consistent tilted axis cranking relativistic mean-field
calculations are performed with configuration reassigned to these bands. The
experimental level schemes of four nuclei are well reproduced, and the
calculations also show four nuclei have obvious triaxial deformations and thus
support the candidate chiral doublet bands in $^{125,127,129,131}$Cs.
