Arrow calculus for welded and classical links

Jean Baptiste Meilhan*, Akira Yasuhara

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


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 finitetype 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.

Original languageEnglish
Pages (from-to)397-456
Number of pages60
JournalAlgebraic and Geometric Topology
Issue number1
Publication statusPublished - 2019 Feb 6
Externally publishedYes

ASJC Scopus subject areas

  • Geometry and Topology


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