Combinatorial link concordance using cut-diagrams

TL;DR

This paper introduces cut-diagrams as a new combinatorial framework for studying link concordance, extending classical notions and establishing invariants like the nilpotent peripheral system.

Contribution

It defines cut-concordance for cut-diagrams and proves that the nilpotent peripheral system is an invariant, providing a combinatorial analogue of Stallings' theorem.

Findings

Nilpotent peripheral system is an invariant of cut-concordance.

Cut-diagrams generalize classical links and surface-links in higher dimensions.

The work connects cut-concordance with existing equivalence relations like link-homotopy.

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.