The invention of the Jones polynomial led to hundreds of papers and a Fields medal. However, as far as I can tell it had few consequences in topology. After all, after Thurston’s work we already had algorithms to completely classify knots, so by itself a new invariant seems to be of limited value. Given all the excitement, however, I suspect that I might be missing something. Why was this so important?



Why should I care about the Jones polynomial?
Your question presupposes that people were excited about the Jones polynomial because it would help them to classify/distinguish knots. In fact, I suspect the interest came from the fact that this knot invariant was originally defined using operator algebras (rather than in the more combinatorial way people usually define it now). Operator algebras had grown out of an attempt to formalise quantum mechanics/QFT, and it was surprising that a knot invariant should appear naturally in a completely different subject. At around the same time, other manifold invariants (Donaldson invariants, instanton Floer homology) appeared that were also inspired by constructions in physics. These other manifold invariants had very definite topological consequences, solving huge open questions in 4-dimensional topology.


Witten showed that all of these invariants (including the Jones polynomial) can be obtained formally by performing path integrals. For example, roughly speaking\*, the Jones polynomial can be obtained by looking at all possible connections A on a suitable bundle on your 3-manifold, taking the trace of the holonomy of A around your knot, multiplying by e^(iCS(A)) where CS is the Chern-Simons invariant of A, and then integrating the result (over the infinite-dimensional space of all connections, which is a kind of path integral). Whether or not you care about physics, that is a pretty cool way to define an invariant.\*\*


This led mathematicians to study topological quantum field theories, and the associated manifold invariants. In particular, Khovanov was led to discover his homological refinement of the Jones polynomial, which is extraordinarily useful as a knot invariant: it is functorial under cobordisms of knots, so can be used to study surfaces in 4-manifolds, for example giving lower bounds on slice genus of knots.


So if you're looking for topological results in pure knot theory that wouldn't exist if it weren't for the Jones polynomial, you should check out Khovanov homology and its applications.


(\*) I corrected what I had written earlier (it's been a long time since I looked at Witten's paper).


(\*\*) The reason I think this is a cool way to define an invariant is that, since the thing you're integrating is covariant under diffeomorphisms, provided your path integral measure is covariant under diffeomorphisms, it would be obvious that you get an invariant. Of course, even if you knew how to define these measures, the devil is in proving that they are invariant (often, classical symmetries of a Lagrangian do not descend to a symmetry of the quantum theory, and the theory is called anomalous).


