Traces on Hecke algebras and the Jones polynomial - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T01:45:38Z http://mathoverflow.net/feeds/question/80853 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/80853/traces-on-hecke-algebras-and-the-jones-polynomial Traces on Hecke algebras and the Jones polynomial Peter Samuelson 2011-11-13T23:15:37Z 2011-11-14T00:36:18Z <p>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 <a href="http://en.wikipedia.org/wiki/Iwahori-Hecke_algebra" rel="nofollow">Iwahori-Hecke algebras</a> \$H(q,n)\$ of type \$A_n\$ to construct the <a href="http://en.wikipedia.org/wiki/HOMFLY_polynomial" rel="nofollow">HOMFLY-PT polynomial</a>, a polynomial invariant of links. In the paper there are a couple statements that are somewhat mysterious:</p> <blockquote> <p>(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.</p> </blockquote> <p><strong>Question</strong>: Have people gone in this direction? Is there a reference?</p> http://mathoverflow.net/questions/80853/traces-on-hecke-algebras-and-the-jones-polynomial/80855#80855 Answer by Jim Humphreys for Traces on Hecke algebras and the Jones polynomial Jim Humphreys 2011-11-14T00:36:18Z 2011-11-14T00:36:18Z <p>The answer to both questions is positive (since mathematicians tend to leave no stone unturned). See for example:</p> <p>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.</p> <p>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.</p>