#### Volume 19, issue 1 (2019)

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Arrow calculus for welded and classical links

### Jean-Baptiste Meilhan and Akira Yasuhara

Algebraic & Geometric Topology 19 (2019) 397–456
##### Abstract

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 $w$–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 Habiro and Shima and to 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 Audoux, Bellingeri and Wagner.

 Dedicated to Professor Shin'ichi Suzuki on his 77th birthday
##### Keywords
knot diagrams, finite-type invariants, Gauss diagrams, claspers
##### Mathematical Subject Classification 2010
Primary: 57M25, 57M27