Sinus-engaging valve fixation member

A prosthesis is provided for implantation at a native semilunar valve of a native valve complex, the native valve complex having three semilunar sinuses and three native commissures. The prosthesis includes a valve prosthesis support, which comprises a support structure comprising exactly three engagement arms that meet one another at three respective junctures. The engagement arms are shaped so as define three peak complexes at the three respective junctures, and three trough complexes, each of which is between two of the peak complexes. Upon implantation of the prosthesis, each of the engagement arms is at least partially disposed within a respective one of the semilunar sinuses, such that each of the peak complexes is disposed distal to and in rotational alignment with a respective one of the native commissures, and each of the trough complexes is disposed at least partially within the respective one of the semilunar sinuses.