Volume 19, issue 1 (2019)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 19
Issue 3, 1079–1618
Issue 2, 533–1078
Issue 1, 1–532

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Subscriptions
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Ethics Statement
Author Index
To Appear
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Other MSP Journals
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
References
Publication
Received: 1 March 2018
Revised: 13 July 2018
Accepted: 10 August 2018
Published: 6 February 2019
Authors
Jean-Baptiste Meilhan
CNRS
Institut Fourier
Université Grenoble Alpes
Grenoble
France
http://www-fourier.ujf-grenoble.fr/~meilhan/
Akira Yasuhara
Faculty of Commerce
Waseda University
Shinjuku-ku
Tokyo
Japan