Arrow calculus for welded and classical links

Jean Baptiste Meilhan, Akira Yasuhara

Research output: Contribution to journalArticle

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 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
Volume19
Issue number1
DOIs
Publication statusPublished - 2019 Feb 6

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Calculus
Diagram
Knot
Gauss
Homotopy
Corollary
Strings
Classify
Higher Order
Analogue
Algebra
Subset
Invariant
Object
Presentation

ASJC Scopus subject areas

  • Geometry and Topology

Cite this

Arrow calculus for welded and classical links. / Meilhan, Jean Baptiste; Yasuhara, Akira.

In: Algebraic and Geometric Topology, Vol. 19, No. 1, 06.02.2019, p. 397-456.

Research output: Contribution to journalArticle

Meilhan, Jean Baptiste ; Yasuhara, Akira. / Arrow calculus for welded and classical links. In: Algebraic and Geometric Topology. 2019 ; Vol. 19, No. 1. pp. 397-456.
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