During the last decade, treatment of aortic valve stenosis using transcatheter aortic valve implantation (TAVI) has been established and is now proven to be at least as good as the traditional surgical treatment in elderly patients and in patients with other diseases besides their cardiac problem. This study, the NOTION-4 trial, is a Nordic collaboration to test the optimal blood thinning treatment strategy after patients have received a transcatheter heart valve. Two medical regimens will be tested in patients with stable heart rhythm (sinus rhythm). In patients with unstable rhythm (atrial fibrillation), who require stronger blood thinning, the effectiveness of implantation of a plug in the heart chamber where most blood clots origin, is tested, with the purpose of reducing the amount of blood thinning and thereby reduce the risk of bleedings and other important side effects. The trial uses the technique of randomization – the preferred scientific technique when testing different treatment strategies.