A graphical calculus for tangles in surfaces

TL;DR

This paper develops a graphical calculus for tangles on surfaces, establishing an equivalence with well-connected tangles and proposing new foundations for link and tangle theory with potential applications in 3-manifolds and state-sum invariants.

Contribution

It introduces a new graphical framework for tangles on surfaces, connecting them with well-connected tangles and enabling novel approaches to invariants.

Findings

Equivalence between tangles and well-connected tangles on surfaces.

Reformulation of graphical foundations for link and tangle theory.

Potential applications to 3-manifold invariants and state-sum methods.

Abstract

We show how the theory of tangles is equivalent to that of well-connected tangles. These are drawn on a surface with boundary, and equivalent via Reidemeister moves of a restricted kind. This reworking of the graphical foundations for link and tangle theory can be expected to have a variety of applications, including ones involving 3-manifolds. It opens the way to new approaches for defining `facial' state-sum invariants that depend in part on assigning substates to faces of tangle diagrams.