Knot a filament and make it active — give each segment the ability to exert force on the surrounding fluid. What happens depends entirely on which knot you tied.
Bonato, Marenduzzo, Orlandini, and Negro simulate active knotted filaments and find that topology programs motility. Torus knots (trefoil, cinquefoil — the ones you can draw on the surface of a torus) inflate: the active forces expand the knot, increasing its spatial extent, and the resulting asymmetric flow field propels the knot through the fluid with a definite chirality. The direction of swimming is determined by the handedness of the knot.
Twist knots (figure-eight, Stevedore — the ones formed by twisting and then closing) do the opposite. They tighten under activity, becoming more compact, and produce no net propulsion. They sit in place, vibrating.
The selection mechanism is geometric. Torus knots have a global chirality — the curve winds consistently in one direction around the torus. When active forces act along such a curve, the resulting flow field has a net dipole component, which drives translation. Twist knots lack this global chirality. Their local twists cancel when integrated over the knot, producing no net dipolar flow. The active force is there, but it generates only quadrupolar or higher-order flow that decays too rapidly to produce propulsion.
The knot behaves as a topological quasi-particle. Its shape, size, and velocity are all determined by the knot type — the topological invariant — rather than by the magnitude of the active force or the details of the fluid. Different activity levels change the speed but not the qualitative behavior. The topology is the program. The activity is the power supply.
Tie a trefoil, and it swims. Tie a figure-eight, and it stays. The instruction set is in the crossing pattern.