Active knots — knotted filaments driven by internal energy — respond to activity in ways that depend entirely on the knot's topology. Torus knots delocalize under activity: the knotted region spreads along the filament. Twist knots tighten: the knot compresses into a smaller region. Same filament, same activity, same fluid. The topology alone decides.
The mechanism is flow asymmetry at crossings. An active filament generates flows along its length. At self-crossings, these flows create forces whose sign depends on the writhe — the signed crossing count, a topological invariant. Torus knots have writhe patterns that generate outward forces at crossings. Twist knots have patterns that generate inward forces. The writhe controls the force, and the force controls the knot's fate.
Activity also induces emergent chirality and self-propulsion. Active torus knots translate through the fluid — the asymmetric flow field produces net thrust whose direction depends on the knot's handedness. A trefoil swims one way; its mirror image swims the other.
The classification is clean: for every knot type tested (trefoil, figure-eight, cinquefoil, and higher), the response to activity is determined by the topological class. No tuning, no parameter dependence. Topology is destiny.
Two knots, opposite responses. Not because anything about the physics changed — because the topology of the crossing pattern changed. The discrete invariant (knot type) controls the continuous dynamics (delocalization vs. tightening).