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(This question is fairly open-ended and probably doesn't have a good answer, so this is more of a suggestion.) One thing that might be useful is a PBW-type basis for $A_q$. In other words, many ``quan …

I'd be interested in doing some computations in quantum groups $ U_q(\mathfrak g)$ that are conceptually simple (``is this element 0"?, and $\mathfrak g = sl_5$), but are somewhat lengthy to do by han …

Here is one reason not to expect such a relationship (although I'm not sure if it can be completed to a proof). The Jones polynomial $J_\sigma$ (roughly) comes from taking the trace of a linear map $A …

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 …

(EDIT: Powers of $q$ in the formula corrected.) I've been doing some computations with skein modules, and I found the following formula for the N-th colored Jones polynomial of the trefoil: $\frac{1 …

The Volume conjecture says that if $J_n(q)$ are the colored Jones polynomials of a knot $K \subset S^3$, then $$\lim_{N \to \infty} \frac{ 2 \pi}N \left\vert J_N(e^{2\pi i / N})\right\vert = vol(K)$$ …