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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 $H(q,n)$ of type $A_n$ to construct the HOMFLY-PT 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.

Question: Have people gone in this direction? Is there a reference?

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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.

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This is a beautiful question that is not trivial at all. After Jones' paper, Lambropoulou constructed a trace on the generalized Hecke algebra of type B, through which, she obtained the analogue of the HOMFLYPT polynomial for knots in the solid torus: http://www.math.ntua.gr/~sofia/publications/A4%20cyclotomic-my%20pdf.pdf

Then, we worked on generalizing this result for knots in Lens spaces Lp,1. Here is a survey on this work: https://arxiv.org/abs/1702.06290

Keep in mind that this survey is not updated, since many more results are now known. I am currently writting an updated survey on skein modules via braids and knot algebras, that I hope will appear soon.

Hope that helps.

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  • $\begingroup$ Let me know when it appears! (I don't check the arXiv as often as I should these days...) $\endgroup$ Commented Oct 27, 2023 at 14:23

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