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2019 Arrow calculus for welded and classical links
Jean-Baptiste Meilhan, Akira Yasuhara
Algebr. Geom. Topol. 19(1): 397-456 (2019). DOI: 10.2140/agt.2019.19.397

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.

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Jean-Baptiste Meilhan. Akira Yasuhara. "Arrow calculus for welded and classical links." Algebr. Geom. Topol. 19 (1) 397 - 456, 2019. https://doi.org/10.2140/agt.2019.19.397

Information

Received: 1 March 2018; Revised: 13 July 2018; Accepted: 10 August 2018; Published: 2019
First available in Project Euclid: 12 February 2019

zbMATH: 07053578
MathSciNet: MR3910585
Digital Object Identifier: 10.2140/agt.2019.19.397

Subjects:
Primary: 57M25, 57M27

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.19 • No. 1 • 2019
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