Questions about quantum invariants such as the Jones, Kauffman, and HOMFLYPT polynomials should have this tag, as should questions about finite type invariants and questions about related universal objects such as the Kontsevich invariant, the KKT invariant, the LMO invariant, Reshetikhin-Turaev invariants, and Turaev-Viro invariants.
Quantum topology begins with the discovery of the Jones polynomial of an oriented knot or link, and its relationships with statistical mechanics and with quantum groups. Edward Witten showed how the Jones polynomial may be recovered physically from Chern-Simons theory on the 3-sphere with gauge group $SU(2)$. By way of singularity theory, Victor Vassiliev discovered the finite-type invariants, placing the Jones polynomial in a broader framework. These are organized by the universal finite-type invariant, that is the Kontsevich invariant. Meanwhile, quantum topological invariants of 3-manifolds such as the Turaev-Viro invariants and the Reshetikhin-Turaev invariants were also discovered.
Quantum topology views its topological subjects as linear combinations of topological objects such as knots, links, and tangles, which are themselves in turn composed of a small collection of basic building which are glued together. Its invariants are typically constructed by assigning a value to each of these basic building blocks, together with a collection of rules, derived from quantum algebra, for how these values are to combine when basic building blocks are glued and modified.
Standard textbooks on quantum topology include Quantum invariants by Ohtsuki, Quantum invariants of knots and 3-maniofolds by Turaev, and Knots and physics by Kauffman.