Burau representation and the Jones polynomial

Question

It is well known that if $K$ is the closure of a braid $f$ in the braid group $B_n$ then, up to some overall factor, the Alexander polynomial $\Delta_K(t)$ of $K$ is given by \begin{equation} \Delta_K(t) = \frac{\det(I-\psi(f))}{1+t+t^2 + ...+t^{n-1}} \ , \end{equation}
where $\psi(f)$ is the Burau representation of $f$. Is there a similar relation involving the Jones polynomial?

Answer 1

On one hand, you seem to be asking if the Jones polynomial of a closure can be given in terms of the Burau representation alone. I think it's pretty easy to find a counterexample to that by finding closures with different Jones polynomials whose underlying braids have the same Burau representation. For instance, we know that the Burau representation is non-faithful for large enough $n$, and Bigelow has given an explicit element in the kernel whose Jones representation is non-trivial (see e.g. Appendix A.3 here)$^1$.

On the other hand, the Burau representation appears naturally as a factor in Jones' Temperley--Lieb algebra representation. A bit of searching gave these recently written lecture notes by Jones -- in Section 5.4 of these, he describes his representation in terms of the action on braids on certain spaces generated by non-crossing pairings of planar points. In his notation, $\beta_{n,n-2}$ is exactly the reduced Burau representation (see Section 7.1).

So in general, to figure out the Jones polynomial of a closure of a braid, you need to figure out the traces of all the representations $\beta_{n,p}$, so what would be ideal to you would be if all the contributions came from $\beta_{n,n-2}$, and in fact, Jones gives a particular example where this is the case in Section 7.5 of those notes: torus knots.

$^1$Edit: Okay, I now got curious and decided to check if that claim is actually true, and according to KnotTheory`, if I didn't make any typos, the value of the Jones polynomial of the closure of Bigelow's element differs in $q = 2$ from that of the closure of the trivial braid:

InvertBraid[arr_] := Reverse[-arr];
psi1 = {-3, 2, 1, 1, 2, 4, 4, 4, 3, 2};
psi2 = {-4, 3, 2, -1, -1, 2, 1, 1, 2, 2, 1, 4, 4, 4, 4, 4};
word1 = Flatten[{InvertBraid[psi1], {4}, psi1}];
word2 = Flatten[{InvertBraid[psi2], {4, 3, 2, 1, 1, 2, 3, 4}, psi2}];
BR2 = Flatten[{word1, word2, InvertBraid[word1],InvertBraid[word2]}];
K1 = BR[5, {}];
K2 = BR[5, BR2];
Jones[K1][2] == Jones[K2][2] (* hopefully returns False *)