  We present an asymptotic theory for solving the dynamics of slender
autophoretic loops and knots. Our formulation is valid for non-intersecting 3D
centrelines, with arbitrary chemical patterning and varying (circular)
cross-sectional radius, allowing a broad class of slender active loops and
knots to be studied. The theory is amenable to closed-form solutions in simpler
cases, allowing us to analytically derive the swimming speed of chemically
patterned tori, and the pumping strength (stresslet) of a uniformly active
slender torus. Using simple numerical solutions of our asymptotic equations, we
then elucidate the behaviour of many exotic active particle geometries, such as
a bumpy uniformly active torus that spins and a Janus trefoil knot, which
rotates as it swims forwards.
