  We show that the map on components from the space of classical long knots to
the n-th stage of its Goodwillie-Weiss embedding calculus tower is a map of
monoids whose target is an abelian group and which is invariant under clasper
surgery. We deduce that this map on components is a finite type-(n-1) knot
invariant. We also compute the second page in total degree zero for the
spectral sequence converging to the components of this tower as Z-modules of
primitive chord diagrams, providing evidence for the conjecture that the tower
is a universal finite-type invariant over the integers. Key to these results is
the development of a group structure on the tower compatible with connect-sum
of knots, which in contrast with the corresponding results for the (weaker)
homology tower requires novel techniques involving operad actions, evaluation
maps, and cosimplicial and subcubical diagrams.
