  We develop a calculus for diagrams of knotted objects. We define Arrow
presentations, which encode the crossing informations of a diagram into arrows
in a way somewhat similar to Gauss diagrams, and more generally w-tree
presentations, which can be seen as `higher order Gauss diagrams'. This Arrow
calculus is used to develop an analogue of Habiro's clasper theory for welded
knotted objects, which contain classical link diagrams as a subset. This
provides a 'realization' of Polyak's algebra of arrow diagrams at the welded
level, and leads to a characterization of finite type invariants of welded
knots and long knots. As a corollary, we recover several topological results
due to K. Habiro and A. Shima and to T. Watanabe on knotted surfaces in
4-space. We also classify welded string links up to homotopy, thus recovering a
result of the first author with B. Audoux, P. Bellingeri and E. Wagner.
