In his famous paper "Hecke algebra representations of braid groups and link polynomials," (Annals 1987), Jones uses a compatible family of traces $tr\_z$ on the [Iwahori-Hecke algebras](http://en.wikipedia.org/wiki/Iwahori-Hecke_algebra) $H(q,n)$ of type $A\_n$ to construct the [HOMFLY-PT polynomial](http://en.wikipedia.org/wiki/HOMFLY_polynomial), a polynomial invariant of links. In the paper there are a couple statements that are somewhat mysterious:



> 
> (pg 336): This might also show how to use the other Hecke algebras (not of type $A\_n$), and their rich representation theory, in some field related to knots...... (pg 343): Other Hecke algebras exist for other Coxeter-Dynkin diagrams and it would be nice to now if any of the ideas of this paper can be suitably modified for them.
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**Question**: Have people gone in this direction? Is there a reference?



Traces on Hecke algebras and the Jones polynomial
The answer to both questions is positive (since mathematicians tend to leave no stone unturned). See for example:


Geck, Meinolf; Lambropoulou, Sofia. Markov traces and knot invariants related to Iwahori-Hecke algebras of type B. J. Reine Angew. Math. 482 (1997), 191–213.


What I don't know offhand is whether there is a useful up-to-date survey of the whole subject area, though I know of several surveys of knot theory.


