As a mathematics graduate student whose research area lies in low-dimensional topology (more precisely, invariants of 3-dimensional topological manifolds), I heard that there exist multiple applications of this theory to theoretical physics, and moreover, that many mathematical problems in the field actually arise from physical ones.
I would appreciate any examples of such applications/motivation, and references to textbooks (or articles which don't require a deep knowledge of physics) where I could read more on the subject.