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Arrow calculus for
welded and classical links
Jean-Baptiste Meilhan and Akira Yasuhara |
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Algebraic & Geometric Topology 19 (2019) 397–456
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Abstract |
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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. |
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Dedicated to Professor Shin'ichi Suzuki on his 77th birthday |
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Keywords
knot diagrams, finite-type invariants, Gauss diagrams,
claspers
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Mathematical Subject Classification 2010
Primary: 57M25, 57M27
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References |
Publication
Received: 1 March 2018
Revised: 13 July 2018
Accepted: 10 August 2018
Published: 6 February 2019
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Authors |
| Jean-Baptiste Meilhan | |
| CNRS Institut Fourier Université Grenoble Alpes Grenoble France |
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| http://www-fourier.ujf-grenoble.fr/~meilhan/ | |
| Akira Yasuhara | |
| Faculty of Commerce Waseda University Shinjuku-ku Tokyo Japan |
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