Arrow calculus for welded and classical links (Q2414168)

A Gauss diagram is a combinatorial object, introduced by \textit{M. Polyak} and \textit{O. Viro} [Int. Math. Res. Not. 1994, No. 11, 445--453 (1994; Zbl 0851.57010)] and \textit{T. Fiedler} [Gauss diagram invariants for knots and links. Dordrecht: Kluwer Academic Publishers (2002; Zbl 1009.57001)], which faithfully encodes $1$-dimensional knotted objects in $3$-space. A result of Goussarov states that any finite-type knot invariant can be expressed as a weighted count of arrow configurations in a Gauss diagram. \par In the paper under review the authors define arrow presentations, which encode the crossing information of a diagram into arrows, and more generally $w$-tree presentations, which can be seen as ``higher-order Gauss diagrams''. The arrow presentations are planar immersed arrows. The authors prove that two such presentations represent equivalent diagrams if and only if they are related by arrow moves. \par 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. They prove that welded string links are classified up to homotopy by welded Milnor invariants. 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. \par As a corollary, the authors recover several topological results due to \textit{K. Habiro} and \textit{A. Shima} [Topology Appl. 111, No. 3, 265--287 (2001; Zbl 0977.57031)] and to \textit{T. Watanabe} [J. Knot Theory Ramifications 15, No. 9, 1163--1199 (2006; Zbl 1121.57010)] on knotted surfaces in $4$-space. They also classify welded string links up to homotopy, thus recovering a result of the first author with Audoux, Bellingeri and Wagner [\textit{B. Audoux} et al., Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 17, No. 2, 713--761 (2017; Zbl 1375.57032)].