I will construct the first poly-time-computable knot polynomial since Alexander's (1928) by using some new commutator-calculus techniques and a Lie algebra $\mathfrak g_1$ which is at the same time solvable and an approximation of the simple Lie algebra $\mathfrak{sl}_2$.
More information can be found here:
http://www.math.toronto.edu/drorbn/Talks/Toronto-1609/