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?

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    $\begingroup$ Yay! I got the 5000th question! :) $\endgroup$ – Sammy Black Apr 1 '10 at 0:38
  • $\begingroup$ I would like to know this too! $\endgroup$ – Hailong Dao Apr 1 '10 at 0:42

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.

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    $\begingroup$ And now there is a reverse trend as well. We would like to see the Jones polynomial realized via some construction that has a more geometric flavor (that doesn't involve knot diagrams). Does this goal have any satisfactory resolution yet? $\endgroup$ – Sammy Black Apr 1 '10 at 0:51
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    $\begingroup$ It's probably better to say Conway rediscovered the skein relation -- it's at the end of section 12 ("Miscellaneous theorems") of Alexander's original paper, but apparently nobody noticed this for decades. $\endgroup$ – Steven Sivek Apr 1 '10 at 1:17
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    $\begingroup$ It's not completely fair to say that people had trouble constructing interesting invariants using homology since most of our understanding of high dimensional topology is obtained from invariants derived from homology, as is a lot of what we know in the classical dimension despite the zoo of QI. (eg high dimensional knot concordance and large classes of high dimensional knots were classified by homological invariants.) Before Jones, knot theory was not about constructing invariants. Now, a big part of it is. That was the change instigated by Jones, then HOMFLY/Witten/Reshetikin-Turaev $\endgroup$ – Paul Apr 1 '10 at 2:32
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    $\begingroup$ @Paul Regarding the second part of your comment, I don't know whether I would completely agree with the characterization of quantum topology as being about constructing invariants. I would say that what we're trying to do is to understand the structure of the set of knot invariants (at least of a certain class), and doing so entails first knowing what they all are, at least in some weak sense. This fits into the idea of understanding a space (in this case, the space of knots, or the space of 3-manifolds) by understanding the space of functions from that space to some simple target space. $\endgroup$ – Daniel Moskovich Apr 1 '10 at 7:49
  • $\begingroup$ @Sammy: I think this paper might be considered a geometric interpretation of the Jones polynomial. front.math.ucdavis.edu/1106.4789 $\endgroup$ – Ian Agol Jun 30 '11 at 15:29

What makes the Jones polynomial dramatic, I think, is not that it is a polynomial invariant per se, but that it came from an unexpected source where nobody had thought of looking. Indeed, there's still no conceptual mathematical explanation for why we should expect knot invariants to come from such a source.
Jones was working on the Potts model in statistical mechanics (how could this possibly be related to topology?). In this context, it was relevant to study representations of the braid group with n strands Bn into the Temperley-Leib algebra TLn. The miracle now is that the Markov trace of the representation of a braid, suitably normalized, is invariant under Markov moves, and is therefore an invariant of the knot obtained by closing the braid. Why should it be a knot invariant, conceptually? Nobody knows.
What makes it even more amazing is that the Jones polynomial turns out to fit into a family with the Alexander polynomial, which is the archetypical algebraic knot invariant, which heuristically suggests that the Jones polynomial is something important which we should be looking at, and which should probably have a sensible topological interpretation.
The Jones polynomial, I think, is "mathematics we can calculate" as opposed to "mathematics we understand", even now, 25 years after its discovery. Yet it turns out to be tremendously powerful, and to have deep connections with other parts of mathematics.

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    $\begingroup$ In terms of the history, I think it's even worse than what you say. Apparently the connection with the braid group only came when someone (Birman?) saw the TL relations, and suggested that it looked a lot like the braid relation; that was enough to extract the representation of $B_n$, and then the trace came. By now there are better explanations of much of this, of course. $\endgroup$ – Dylan Thurston Apr 1 '10 at 5:43
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    $\begingroup$ Vaughan Jones was working on subfactors. He discovered the Temperley-Lieb algebras independently. The fact that the algebras Vaughan Jones used to construct subfactors with index at most 4 and the algebras Temperley-Lieb introduced in order to show that the six vertex model and the Potts model had the same partition function was pointed out by David Evans. I have also heard a quip that the Jones polynomial should be called Joan's polynomial. I don't know if this is just being clever or whether there is some justification. $\endgroup$ – Bruce Westbury Apr 1 '10 at 6:05
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    $\begingroup$ I have only heard this second hand, but it is said that Jones offered Joan Birman coauthorship. $\endgroup$ – Scott Carter Apr 1 '10 at 12:57
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    $\begingroup$ A good source for Joan Birman's part in the Jones polynomial story is her interview in the Notices. In particular, according to that interview it was Vaughan who noticed the connection to the braid group which is why he went to talk to Joan. She then explained to him about Markov relations and then they had a miscommunication about "matrix trace" vs. "multi-matrix trace" which resulted in them not figuring out what was going on during their in-person discussion. Vaughan then separately figured out what was going on. They then did the first computations together. $\endgroup$ – Noah Snyder Apr 3 '10 at 4:27
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    $\begingroup$ For another primary source, Vaughan's paper "A polynomial invariant for knots via von Neumann algebras" says "The author would like to thank Joan Birman. It was because of a long discussion with her that the relation between condition (V) and Markov's theorem became clear." $\endgroup$ – Noah Snyder Apr 3 '10 at 4:30

I think that V.F.R. Jones told me that he originally thought that it was a variation of the Alexander polynomial. According to J.H.Conway, Lickorish and Millett were working on their version of HOMFLYPT under Conway's nose at Cambridge and not telling him a thing about it. Another thing that Vaughn mentioned was that in the 1980s not many people were looking directly at the braid group until the discovery of the Jones polynomial. This is more or less true. Ralph Cohen certainly was, but in the context of $\Omega^2 S^3.$

Specifically, in low dimensional topology there was, naturally, more focus on Bill Thurston's work and the work of his disciples.

In relation to your follow-up to Dylan's answer, many people would argue that Witten's formulation of the Jones polynomial is a topological definition.

It seems that a more reasonable answer is yet to come in the form of identifying where the pieces of Khovanov homology come from.

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    $\begingroup$ Joan Birman is another notable example of someone looking directly at the braid group prior to the Jones polynomial. $\endgroup$ – Noah Snyder Apr 1 '10 at 5:35
  • $\begingroup$ Yes, indeed! Vaughn told me directly that Joan's interest in braids at this point in history was waning. $\endgroup$ – Scott Carter Apr 1 '10 at 12:58

You can find some historical remarks on polynomial invariants in chapter 9 of the book "Knots and links" by Cromwell, he also gives a lot of references.


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