  Atom optics demonstrates optical phenomena with coherent matter waves,
providing a foundational connection between light and matter. Significant
advances in optics have followed the realisation of structured light fields
hosting complex singularities and topologically non-trivial characteristics.
However, analogous studies are still in their infancy in the field of atom
optics. Here, we investigate and experimentally create knotted quantum
wavefunctions in spinor Bose--Einstein condensates which display non-trivial
topologies. In our work we construct coordinated orbital and spin rotations of
the atomic wavefunction, engineering a variety of discrete symmetries in the
combined spin and orbital degrees of freedom. The structured wavefunctions that
we create map to the surface of a torus to form torus knots, M\"obius strips,
and a twice-linked Solomon's knot. In this paper we demonstrate striking
connections between the symmetries and underlying topologies of multicomponent
atomic systems and of vector optical fields--a realization of topological
atom-optics.
