Algebraic proof of 4-colour theorem? 4-colour Theorem. Every planar graph is 4-colourable.
This theorem of course has a well-known history.  It was first proven by Appel and Haken in 1976, but their proof was met with skepticism because it heavily relied on the use of computers.  The situation was partially remedied 20 years later, when Robertson, Sanders, Seymour, and Thomas published a new proof of the theorem.  This new proof still relied on computer analysis, but to such a lower extent that their proof was actually verifiable. Finally, in 2005, Gonthier and Werner used the Coq proof assistant to formalize a proof, so I suppose only the most die hard skeptics remain.
My question stems from reading An Update on the Four-Color Theorem by Robin Thomas.  The paper describes several interesting reformulations of the 4-colour theorem.  Here is one:
Note that the cross-product on vectors in $\mathbb{R}^3$ is not an associative operation.  We therefore define a bracketing of a cross-product $v_1 \times \dots v_n$ to be a set of brackets which makes the product well-defined.
Theorem. Let $i, j, k$ be the standard unit vectors in $\mathbb{R}^3$.  For any two different bracketings of the product $v_1 \times \dots \times v_n$, there is an assignment of $i,j,k$ to $v_1, \dots, v_n$ such that the two products are equal and non-zero.
The surprising fact is that this innocent looking theorem implies the 4-colour theorem.
Question. Is anyone working on an algebraic proof of the 4-colour theorem (say by trying to prove the above theorem)?  If so, what techniques are involved?  What partial progress has been made?  Or do most people consider the effort/reward ratio of such an endeavor to be too high?
I think it would be interesting to have an algebraic proof, even a very long one, particularly if the algebraic proof does not use computers. Given its connection to many other areas (Temperley-Lieb Algebras), the problem seems to be amenable to other forms of attack.
 A: There is an algebraic method by Alon and Tarsi which allows in certain cases to prove that certain graphs are $k$-colorable (in fact, even $k$-choosable). A famous case where this method prevails is to show that a graph on $3n$ vertices consist of edge-disjoint union of a Hamiltonian cycle and $n$ triangles is 3-colorable. This method can potentially show that graphs of cubic planar bridgeless graphs are 3-edge colorable which is equivalent to the 4CT. (So far it allows only to derive 3-edge choosability from the 4CT.) 
(I suppose that various other speculative algebraic methods were proposed over the years. For example, I myself proposed to approach 4CT by finding algebraic/topological conditions for the existence of "Tverberg's partitions.)   
A: There are several reformulations by Matiyasevich, like this with polynomial, this with binomial coefficients modulo 7, also some others but they are less `algebraic', whatever it means.
A: I must admit I'm a bit baffled about what the question is here, and about why so many people have voted it up.  What are you looking for in an answer?  I don't think it's appropriate to post speculation on the internet about which mathematicians are privately working on which big problems.  As to public work, you seem to have a weirdly restrictive view of what "working on" and "partial progress" mean that don't fit with my understanding of how mathematics works.  Several papers have been written on the subject of possible algebraic proofs of the 4-color theorem (look at google scholar or Mathscinet for papers which cite the Saleur-Kauffman paper mentioned in the paper you're reading), but if the Bar-Natan paper doesn't count for you then you're likely to be disappointed by all of them.
The long and short of it is that everyone in quantum topology would love to prove the 4-color theorem and occasionally thinks about it.  There's lots of tantalizing clues that an algebraic argument has promise, but if anyone knew how to prove it they'd have done so.  As far as I know there isn't anyone who is holed up in their attic thinking about only the 4-color theorem, instead there's a lot of people who every time they find a new tool think "hrm, I wonder if this tool would work on the 4-color theorem?"
A: There is a classical approach by Birkhoff and Lewis, which remained dormant for decades.  It was recently revived by Cautis and Jackson (start here [“The matrix of chromatic joins and the Temperley-Lieb algebra”, J. Combin. Theory 89 (2003), 109–155] and proceed here [“On Tutte's chromatic invariant”, Trans. Amer. Math. Soc. 362 (2010), 491–507]), using the Temperley-Lieb algebra.
A: Does "Lie Algebras and the Four Color Theorem" (Combinatorica 17, 43–52 (1997). https://doi.org/10.1007/BF01196130) by Dror Bar-Natan qualify ?
