Quantum Groups and Knot Invariants

The Jones polynomial $V_K(q) \in \mathbb{Z}[q^{\pm 1/2}]$ revolutionized knot theory when it was discovered \in 1984. The representation-theoretic explanation came from quantum groups: the Drinfeld-Jimbo deformation $U_q(\mathfrak{sl}_2)$ carries a universal $R$-matrix satisfying the Yang-Baxter equation, and representations of $U_q(\mathfrak{g})$ produce a systematic family of knot invariants via the Reshetikhin-Turaev construction. We trace this story from the skein relation to the HOMFLY-PT polynomial, then discuss Khovanov's categorification, which lifts $V_K(q)$ to a bigraded homology theory $\text{Kh}(K)$ whose graded Euler characteristic recovers the Jones polynomial.