Three-quasiparticle K-isomeric states in odd-mass N = 106 isotones within the \(A\,\sim\, 180\) mass region were systematically investigated using configuration-constrained potential energy surface calculations. The calculations successfully reproduced the excitation energies and deformations of the known high-K isomers in nuclei from 175Tm to 181Re. For the nuclei closer to the \(Z=82\) shell closure (183Ir, 185Au, and 187Tl), predictions of the configurations of the observed and yet-to-be-observed isomers are provided. The results reveal strong shape polarization, where the three-quasiparticle states are driven to larger deformations compared to the often shape-soft or spherical ground states. A particularly rich spectrum of shape coexistence is predicted in 187Tl, where several high-K three-quasiparticle configurations with distinct prolate, oblate, and triaxial shapes are found to coexist at similar excitation energies. Notably, the oblate-deformed \(K^{\pi }=29/2^{+}\) configuration at \(E_x = 1839\) keV was proposed to be responsible for a long-lived isomer. This study provides a comprehensive picture of shape evolution and coexistence in high-K multi-quasiparticle states, offering valuable insights for future experimental studies.
