Knot invariants such as the Jones and HOMFLYPT polynomials, and their categorifications, are central tools in low-dimensional topology. But how effective are they as classifiers? In this talk I will explain a recent work with A. Lacabanne, D. Tubbenhauer, and V. Zhang, where we show that many classical and quantum invariants detect alternating links with probability zero: their distinguishing power decays exponentially due to insensitivity to oriented mutation. Large-scale computations further reveal that categorified invariants perform no better than their polynomial counterparts, highlighting intrinsic limits of knot invariants and pointing to new challenges in quantum topology.