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We develop a calculus for diagrams of knotted objects. We define Arrowpresentations, which encode the crossing informations of a diagram into arrowsin a way somewhat similar to Gauss diagrams, and more generally w-treepresentations, which can be seen as `higher order Gauss diagrams'. This Arrowcalculus is used to develop an analogue of Habiro's clasper theory for weldedknotted objects, which contain classical link diagrams as a subset. Thisprovides a 'realization' of Polyak's algebra of arrow diagrams at the weldedlevel, and leads to a characterization of finite type invariants of weldedknots and long knots. As a corollary, we recover several topological resultsdue to K. Habiro and A. Shima and to T. Watanabe on knotted surfaces in4-space. We also classify welded string links up to homotopy, thus recovering aresult of the first author with B. Audoux, P. Bellingeri and E. Wagner.

Arrow calculus for welded and classical links