Filamentous structures are abundant in cells. Relatively rigid filaments, such as microtubules and actin, serve as intracellular scaffolds that support movement and force, and their mechanical properties are crucial to their function in the cell. Some aspects of the behaviour of DNA, meanwhile, depend critically on its flexibility-for example, DNA-binding proteins can induce sharp bends in the helix. The mechanical characterization of such filaments has generally been conducted without controlling the filament shape, by the observation of thermal motions or of the response to external forces or flows. Controlled buckling of a microtubule has been reported, but the analysis of the buckled shape was complicated. Here we report the continuous control of the radius of curvature of a molecular strand by tying a knot in it, using optical tweezers to manipulate the strand's ends. We find that actin filaments break at the knot when the knot diameter falls below 0.4 microm. The pulling force at breakage is around 1 pN, two orders of magnitude smaller than the tensile stress of a straight filament. The flexural rigidity of the filament remained unchanged down to this diameter. We have also knotted a single DNA molecule, opening up the possibility of studying curvature-dependent interactions with associated proteins. We find that the knotted DNA is stronger than actin.