Algebraic Knots and their Universal K^MW_2-Coverings

Over suitable base fields $k$ of characteristic not $2$, including algebraically closed ones, we construct universal abelian ${\underset{}{\mathrm{K}}}_2^{\mathrm{MW}}$-coverings for complements of closed embeddings ${\mathbb{A}}^1\hookrightarrow {\mathbb{A}}^3$. Using these, we obtain a rectifiability invariant of such embeddings by lifting knot-theoretic ideas to algebraic geometry via motivic homotopy theory.