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Arrow calculus for welded and classical links

Jean-Baptiste Meilhan and Akira Yasuhara

Algebraic & Geometric Topology 19 (2019) 397–456

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

knot diagrams, finite-type invariants, Gauss diagrams, claspers
Mathematical Subject Classification 2010
Primary: 57M25, 57M27
Received: 1 March 2018
Revised: 13 July 2018
Accepted: 10 August 2018
Published: 6 February 2019
Jean-Baptiste Meilhan
Institut Fourier
Université Grenoble Alpes
Akira Yasuhara
Faculty of Commerce
Waseda University