## 1 Introduction  
Dielectric nanostructures at optical frequencies are characterized by an extremely complex landscape of interacting resonant states. By finely tuning the material and geometrical properties of the nanostructures, it is possible to engineer advanced functionalities and applications such as, e.g., anti-reflection surfaces [1] and integrated waveguides based on chains of nanoparticles [2]. One of the most fascinating manifestations of multi-mode interaction in dielectric nanoparticles is the formation of radiationless states known as anapoles. These states have recently been demonstrated in silicon nanoparticles [3]. Anapoles are characterized by a strong reduction of the scattering from the nanoparticle at the anapole wavelength, which acquires the character of a fully-cloaked structure [4; 5]. The mechanism underlying the formation of anapole states is the superposition of internal multi-mode components of the nanoparticle, which cancel each other in the far-field and which produce a radiation pattern confined to the near-field. As a result, anapole states are not the result of any resonant process (as in the case of dark resonances), but their origin lies entirely in modal competition and superposition [6]. While the theoretical description of anapole states in single dielectric nanoparticles is well established, the mutual coupling and interaction along a chain of anapole nanoparticles have yet to be investigated. Arrays of nanoparticles have been the subject of intense study, starting from the remarkable guiding properties demonstrated in plasmonic nanochains [7; 8]. To mitigate the effect of metal losses, researchers have focused on all-dielectric solutions based on high refractive index nanoparticles [9]. As discussed in [10], the guiding properties of nanoparticle arrays can be strongly enhanced by minimizing the electromagnetic scattering from each nanoparticle in the array. As a result, anapole states could represent--in principle--a perfect candidate for efficient integrated waveguides based on silicon nanoparticles. However, due to their intrinsic non-resonant nature, anapoles should not manifest mutual coupling (as would be expected in the case of standard resonant states). Therefore, it should not be possible to transfer radiationless states among closely-packed dielectric structures. As recently demonstrated in two-dimensional cylinders, however, anapole states can be re-interpreted as the result of Fano interference between two or more, overlapping resonant states in the proximity of the anapole wavelength [11]. If such analysis could be extended to the three-dimensional case, it would be possible to describe and tailor the mutual coupling of an ensemble of invisible nanostructures. In this work, we address this problem by combining first-principles simulations and analytical theory, showing how the anapole state can be effectively transferred among distinct nanoparticles. Our results play a key role in the development of integrated optical circuitry based on non-radiating states (e.g., anapole-based wave-guides). Due to the near-field confinement produced by the anapole excitation and the suppression of the scattered field, wave-guides based on non-radiating states are extremely robust against physical bending and splitting, opening to the realization of high-density optical circuitry entirely based on silicon.