Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On finite type invariants of welded string links and ribbon tubes

Adrien Casejuane, Jean-Baptiste Meilhan

TL;DR

The paper develops a finite type invariant theory for welded string links via arrow calculus and $w_k$-equivalence, connecting combinatorial welded diagrams to ribbon knotted surfaces in 4-space through the Tube map. It provides explicit low-degree classifications: degree-1 linking numbers, degree-2 closure and Milnor-type invariants suffice to classify welded string links up to $w_3$-equivalence, and a partial $w_4$-classification is outlined for the two-component case. The results yield concrete decompositions of welded string links into products of elementary generators, with precise invariant-based coefficients, and confirm that degree-2 invariants are generated by closure invariants while degree-3 invariants require additional Milnor data. These diagrammatic classifications translate to topological consequences for ribbon tubes via the Tube map, giving RC$_k$-type classifications and showing that the relevant groups are finitely generated and largely abelian only in trivial cases. Overall, the work advances Goussarov–Habiro-style classifications in higher dimensions and provides tools to study ribbon knotted surfaces through welded combinatorics and arrow calculus.

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.

On finite type invariants of welded string links and ribbon tubes

TL;DR

The paper develops a finite type invariant theory for welded string links via arrow calculus and -equivalence, connecting combinatorial welded diagrams to ribbon knotted surfaces in 4-space through the Tube map. It provides explicit low-degree classifications: degree-1 linking numbers, degree-2 closure and Milnor-type invariants suffice to classify welded string links up to -equivalence, and a partial -classification is outlined for the two-component case. The results yield concrete decompositions of welded string links into products of elementary generators, with precise invariant-based coefficients, and confirm that degree-2 invariants are generated by closure invariants while degree-3 invariants require additional Milnor data. These diagrammatic classifications translate to topological consequences for ribbon tubes via the Tube map, giving RC-type classifications and showing that the relevant groups are finitely generated and largely abelian only in trivial cases. Overall, the work advances Goussarov–Habiro-style classifications in higher dimensions and provides tools to study ribbon knotted surfaces through welded combinatorics and arrow calculus.

Abstract

Welded knotted objects are a combinatorial extension of knot theory, which can be used as a tool for studying ribbon surfaces in -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 -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 -equivalence. All results have direct corollaries for ribbon knotted surfaces.
Paper Structure (16 sections, 20 theorems, 35 equations, 3 figures)

This paper contains 16 sections, 20 theorems, 35 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Welded objects and arrow calculus
  3. Welded knotted objects
  4. Arrow calculus
  5. Invariants of welded string links
  6. Closure invariants
  7. Welded linking numbers and Milnor invariants
  8. Finite type invariants
  9. $w_k$-equivalence
  10. Definition and relation to finite type invariants
  11. Refined arrow calculus
  12. The group of welded string links up to $w_k$-equivalence
  13. Characterization of low degree invariants of welded string links
  14. Classification of welded string links up to $w_3$-equivalence
  15. Towards a $w_4$-classification of welded string links
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $T$ and $T'$ be two $n$-component ribbon tubes. The following are equivalent for $k \in \{2;3\}$.

Figures (3)

  • Figure 2.1: The Detour and OC moves (In the Detour move, the grey part indicates any subdiagram, with classical and/or virtual crossings)
  • Figure 2.2: The closures $Cl_{(2,\overline{1})}(L)$ (left) and $Cl_{(1,2)}(L)$ (right) of the welded string link $L$.
  • Figure 3.1: The welded string links $D$ and $D'$, and the welded long knot $K$

Theorems & Definitions (56)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • Theorem 2.9
  • ...and 46 more