We develop a calculus for diagrams of knotted objects. We define arrow
presentations, which encode the crossing information of a diagram into
arrows in a way somewhat similar to Gauss diagrams, and more generally
–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 Habiro and Shima and to Watanabe on knotted surfaces in
–space.
We also classify welded string links up to homotopy, thus recovering a result of the
first author with Audoux, Bellingeri and Wagner.
Dedicated to Professor Shin'ichi
Suzuki on his 77th birthday
