>>11425260Indeed they can. Given a link , we can form a 2-fold by gluing some collection of discs to it, whose world-volume is a 3-manifold classified topologically (surgery theory) by modulo ambient isotopy and Kirby moves. Just speaking in terms of the underlying TQFT, which maps to a state vector in the topological theory, the ground state formed this way define a topological vacuum labeled by link invariants ; in addition, cobordant links define unitarily
equivalent ground states (in the TQFT).
In the string theory itself, the ground states in fact depend on more intricate geometric structures on , such as the conformal class of the metric on it, but this is determined by much more than just the embedded links inside.
