Following Rozansky~\cite{Rozansky:Flat1,Rozansky:Burau,Rozansky:U1RCC}
and Overbay~\cite{Overbay:Thesis}, we construct
the first poly-time-computable knot polynomials since
Alexander's~\cite[1928]{Alexander:TopologicalInvariants}. We use
some new commutator-calculus techniques and a family of Lie algebra
$sl_2^{\leq k}$ which are solvable yet at the same time they make progressively
better approximations of the simple Lie algebra $sl_2$. The resulting
invariants are the strongest genuinely-computable knot invariants
presently available and they seem to contain information about some
classical topologically-defined knot invariants.