  We construct and prove a diagrammatic version of the Duflo isomorphism
between the invariant subalgebra of the symmetric algebra of a Lie algebra and
the center of the universal enveloping algebra. This version implies the
original for metrized Lie algebras (Lie algebras with an invariant
non-degenerate bilinear form). As an application of this isomorphism, we will
compute the Kontsevich integral of the unknot and the Hopf link to all orders.
  At the core of the proof, we use an elementary property of the Hopf link
which can be summarized by the equation ``1+1=2'' in abacus arithmetic:
doubling one component of the Hopf link is equivalent to taking the connected
sum of two Hopf links. This property of the Hopf link turns out, when suitably
interpreted, to be exactly the property required for the Duflo map to be
multiplicative.
  To compute the Kontsevich integral of the unknot, we use a property of the
unknot that can be summarized by ``n * 0 = 0'': the n-fold connected cabling of
the unknot is again an unknot.
