I apologize that this is vague, but I'm trying to understand a little bit of the historical context in which the zoo of quantum invariants emerged.
For some reason, I have in my head the folklore:
The discovery in the 80s by Jones of his new knot polynomial was a shock because people thought that the Alexander polynomial was the only knot invariant of its kind (involving a skein relation, taking values in a polynomial ring, ??). Before Jones, there were independent discoveries of invariants that each boiled down to the Alexander polynomial, possibly after some normalization.
Is there any truth to this? Where is this written?
Best Answer
The skein relation approach to knot invariants was not very popular before the Jones polynomial. The Alexander polynomial was thought of as coming from homology (of the cyclic branched cover); Conway had found the skein relation, but it was not well-known. Of course once you start investigating skein relations systematically, you rapidly find the Jones, Kauffman, and HOMFLY relations.
Basically, people had been looking for invariants using their standard tools like homology, and had trouble constructing interesting invariants that way. The idea of just looking for a skein relation was new. The notion of "polynomial invariants" by itself is too vague to give a place to look.