  This paper determines the full derived deformation theory of certain smooth
rational curves C in Calabi-Yau 3-folds, by determining all higher
A_\infty-products in its controlling DG-algebra. This geometric setup includes
very general cases where C does not contract, cases where the curve
neighbourhood is not rational, all known simple smooth 3-fold flops, and all
known divisorial contractions to curves. As a corollary, it is shown that the
noncommutative deformation theory of C can be described as a superpotential
algebra derived from what we call free necklace polynomials, which are elements
in the free algebra obtained via a closed formula from combinatorial gluing
data. The description of these polynomials, together with the above results,
establishes a suitably interpreted string theory prediction due to Ferrari,
Aspinwall-Katz and Curto-Morrison. Perhaps most significantly, the main results
give both the language and evidence to finally formulate new contractibility
conjectures for rational curves in CY 3-folds, which lift Artin's celebrated
results from surfaces.
