Inspired by the work of Michor et al. on the manifold of immersed curves, we propose a new strong Riemannian metric on the manifold of (parametrized) embedded curves of $H^s$-regularity, $s\in (3/2,2)$. The construction is motivated by the concept of tangent-point energies, a family of self-avoiding functionals on curves and surfaces of arbitrary dimension. Their notion of self-repulsion allowed us to capture the topological property ”embeddedness” in a continuous way. This talk illustrates the impact of this metric and highlights its most important features, namely metric and geodesic completeness, relative compactness of bounded sets with respect to the weak $H^s$-topology, and existence of length-minimizing geodesics between every pair of curves in the same knot class. This is joint work with Philipp Reiter and Henrik Schumacher.