Combinatorial link concordance using cut-diagrams

Abstract

Cut-diagrams are diagrammatic objects, defined in dimensions 1 and 2, that generalize links in 3-space and surface-links in 4-space; in dimension 1, this coincides with the theory of welded links. Using cut-diagrams, we introduce an equivalence relation called cut-concordance, which encompasses the topological notion of concordance for classical links. Our main result is that the nilpotent peripheral system of 1--dimensional cut-diagrams is an invariant of cut-concordance, giving along the way a combinatorial version of a theorem of Stallings. We also investigate the relationship with several other equivalence relations in diagrammatic knot theory, in particular in connection with link-homotopy.

Figures (14)

• Figure 1: $1$--dimensional cut-diagrams arising from tangle diagrams. Regions are named with capital letters, and labels on cut-points are given by circled nametags
• Figure 2: Topological moves on $1$--dimensional cut-diagrams ($\varepsilon,\eta=\pm$): in the middle and right moves, region $D$ must not occur as the label of any cut-point; in the left/middle move, when going from left to right, every $A$--label becomes either an $A$ or $B$--label, and when going from right to left, all the $A$ and $B$--labels become $A$--labels
• Figure 3: The mixed and OC moves
• Figure 4: Labeling rules for $2$--dimensional cut-diagrams
• Figure 5: Local models for triple points and branch points in a surface diagram (left), and the associated local cut-diagrams (right)

Theorems & Definitions (47)

• Theorem : Thm. \ref{['th:NilpotentConcordance']}
• Definition 1.1
• Remark 1.2
• Proposition 1.3
• Definition 1.4
• Proposition 2.1
• proof
• Proposition 2.2
• Definition 2.3
• Remark 2.4