On finite type invariants of welded string links and ribbon tubes

TL;DR

This paper explores finite type invariants of welded string links and ribbon tubes, establishing their algebraic structure and how they characterize invariants in low degrees, with implications for ribbon knotted surfaces.

Contribution

It introduces a detailed study of welded string links up to w_k-equivalence, connecting finite type invariants with algebraic structures and ribbon surface theory.

Findings

w_k-equivalence characterizes finite type invariants in low degrees

algebraic structures of welded string links are elucidated

results have direct implications for ribbon knotted surfaces

Abstract

Welded knotted objects are a combinatorial extension of knot theory, which can be used as a tool for studying ribbon surfaces in $4$-space. A finite type invariant theory for ribbon knotted surfaces was developped by Kanenobu, Habiro and Shima, and this paper proposes a study of these invariants, using welded objects. Specifically, we study welded string links up to $w_k$-equivalence, which is an equivalence relation introduced by Yasuhara and the second author in connection with finite type theory. In low degrees, we show that this relation characterizes the information contained by finite type invariants. We also study the algebraic structure of welded string links up to $w_k$-equivalence. All results have direct corollaries for ribbon knotted surfaces.