Quantum Invariants of 3-Manifolds and Links: A Survey

We survey the recent developments on quantum invariants of 3-manifolds and links: Z^ and FL. They are q-series invariants originated from mathematical physics, inspired by the categorification of a numerical quantum invariant—the Witten–Reshetikhin–Turaev (WRT) invariant—of 3-manifolds. They exhibit rich features, for example, quantum modularity, infinite-dimensional Verma module structures, and knot–quiver correspondence. Furthermore, they have connections to the 3d-3d correspondence and other topological invariants. We also provide a review of an extension of the above series invariants to Lie superalgebras.