Even with the connection to quantum groups being made clearer (I believe it was not known when the Jones polynomial was first introduced), it seems to me that still we don't have the "right" definition of the Jones polynomial. It is certainly true though that we know a lot of different definitions, some more useful than others. I'm thinking of:

1) Kauffman bracket (i.e. the skein relation). This defines the Jones polynomial by giving a straightforward algorithm to compute it, but leaves any other significance a mystery.

2) Quantum groups. Take $U_q(\mathfrak s\mathfrak l_2)$, and observe via universal $R$-matrices that "its category of representations is a braided monoidal tensor category", so that in particular, $\overbrace{V\otimes\cdots\otimes V}^{n\text{ times}}$ gives a representation of $B_n$, for any given representation $V$ of $U_q(\mathfrak s\mathfrak l_2)$. The Jones polynomial is easily derived from this representation of $B_n$.

3) KZ equations (closely related to (2)). Let $X_n$ be the configuration space of $n$ points $(z_1,\ldots,z_n)$ in $\mathbb C$. Now write down the one-form $A=\hbar\cdot\sum_{i<j}\Omega_{ij}d\log(z_i-z_j)$ (taking values in $U(\mathfrak s\mathfrak l_2)^{\otimes n}$), and observe that this gives a flat connection on a trivial bundle of $V^{\otimes n}$ over $X_n$, for a representation $V$ of $\mathfrak s\mathfrak l_2$. The monodromy of this connection gives a representation of $\pi_1(X_n)=B_n$ on $V^{\otimes n}$.

Methods (2) & (3) (especially method 3) are natural constructions for representations of $B_n$.

Question: Are there any other constructions of the Jones polynomial that are not trivially (interpret as you wish) equivalent to the ones above?

I am particularly interested in ones which seem natural for the case of *knots in $\mathbb R^3$* (note that (2) and (3) seem natural ways to get representations of $B_n$, but, at least to me, it seems that the extension to knots is sort of ad-hoc). I feel like there are a number of "moral" approaches which "should" give the Jones polynomial, but have yet to be made rigoruous, and I'd be interested to know how close they are to being so:

A) [warning: this is kind of sketchy] Start with $M_K=\operatorname{Hom}(\pi_1(\mathbb S^3-K),G)/\\!/G$ (where $G=\operatorname{SL}(2)$) and consider this as a left-module over $R=\operatorname{Hom}(\pi_1(\text{torus}),G)/\\!/G$. Make a noncommutative deformation $R^q$ of $R$ to get the Kauffman bracket skein module of the torus, and observe that $M_T^q$, the Kauffman bracket skein module of the solid torus $D^2\times S^1$ is a right-module over $R^q$. Since the Kauffman bracket skein module of $\mathbb S^3$ is $\mathbb C$, this means $M_K^q\otimes_{R^q}M_T^q=\mathbb C$. Then take $1\in M_K^q$ and some canonical elements (Jones-Wenzel idempotents) in $M_T^q$ and take their tensor in $M_K^q\otimes_{R^q}M_T^q=\mathbb C$. This should give the colored Jones polynomial of the knot. The problem with this is that we don't know how to define the deformed left-module structure on $M_K$ to get $M_K^q$.

B) Take an ideal triangulation of the knot complement. Apply some black magic "TQFT with corners" (perhaps just some explicit formulae) and get back the Jones polynomial of the knot. I thought that Dylan Thurston was working on this at one point (in relation to the volume conjecture), but that was a while ago, and as far as I know, there is still no definition of the Jones polynomial from an ideal triangulation of the complement (I'm thinking something along the lines of Turaev-Viro invariants of $3$-manifolds). (Please correct me if I'm wrong)

(Certainly my question could be stated in a more general setting for general quantum knot invariants.)

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