We report on a new phenomenon of light guidance in a fiber core created by an arrangement of holes making a partially open ring. In such structure there is no complete refractive index barrier to confine the light and therefore it cannot guide light if untwisted. However, if the open ring of holes is shifted off the symmetry axis of the twisted fiber then the mode confinement and low loss propagation is possible due to purely geometrical effect related to the increase in the optical path of light following the helical route. Properties of such structures, including confinement loss, modal field distribution, birefringence of fundamental modes were investigated both numerically and experimentally. We also studied the effect of bending, which leads to periodic modulation of the propagation characteristics of the twisted fiber. Furthermore, we demonstrate a possibility of displacement measurements based on bend-induced loss using some of the fabricated fibers, in which the sensitivity to bend is controlled by a twist rate and core structure.