Arrow calculus for welded and classical links

Abstract

We develop a calculus for diagrams of knotted objects. We define Arrow presentations, which encode the crossing informations 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 K. Habiro and A. Shima and to T. 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 B. Audoux, P. Bellingeri and E. Wagner.

40 pages, with many figures ; v.2: exposition revised, minor changes

References in corpus (2)

• On codimension two embeddings up to link-homotopy
• On the Kontsevich integral of Brunnian links

Cited by in corpus (11)

• On codimension two embeddings up to link-homotopy
• Characterization of the reduced peripheral system of links
• Milnor invariants of braids and welded braids up to homotopy
• Classification of string links up to $2n$-moves and link-homotopy
• Link invariants derived from multiplexing of crossings
• Generalized virtualization on welded links
• Combinatorial approach to Milnor invariants of welded links
• On finite type invariants of welded string links and ribbon tubes
• Homotopy braid groups are torsion-free
• Goussarov-Polyak-Viro's $n$-equivalence and the pure virtual braid group
• Milnor invariants, $2n$-moves and $V^n$-moves for welded string links