The Whitney move is a procedure for eliminating singularities
among submanifolds of complementary dimension via a homotopy
supported near a 2-dimensional Whitney disk. In dimensions greater
than or equal to 5, the existence of disjointly embedded Whitney
disks follows from algebraic data and general position, and the
Whitney move plays a key role in many classification and
unknotting theorems for high-dimensional manifolds. The failure of
the Whitney move in dimensions less than or equal to 4 is a
defining characteristic of Low-dimensional Topology which
significantly complicates applications of algebraic topology and
surgery theory, and leads to such phenomena as exotic 4-spaces and
intersection forms which are realized by topological but not
smooth closed 4-manifolds. I will describe a combinatorial
approach to measuring the failure of the Whitney move in terms of
higher order intersection trees, which are related to the
Kontsevich integral and Lie algebras of iterated commutators.