Arrow calculus for welded and classical links

TL;DR

This paper introduces an arrow calculus for welded and classical links, enabling the characterization of finite type invariants and recovering several topological results in knot theory and knotted surfaces.

Contribution

It develops a new arrow calculus framework that generalizes Gauss diagrams and extends Habiro's clasper theory to welded knotted objects, linking algebraic and topological invariants.

Findings

Characterization of finite type invariants of welded knots and long knots

Recovery of topological results on knotted surfaces in 4-space

Classification of welded string links up to homotopy

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.