Manifold diagrams and tame tangles

TL;DR

This paper introduces manifold diagrams and tame tangles as higher-dimensional diagrammatic tools, providing a combinatorial classification and exploring their stability and applications in modeling smooth manifolds.

Contribution

It formally defines manifold diagrams and tame tangles, classifies them combinatorially, and demonstrates their relevance to modeling smooth manifolds in higher dimensions.

Findings

Manifold diagrams and tame tangles admit a combinatorial classification.

Tame tangles are stable under perturbations, modeling differential singularities.

All smooth 4-manifolds can be represented as tame tangles.

Abstract

Diagrammatic notation has become a ubiquitous computational tool; early examples include Penrose's graphical notation for tensor calculus, Feynman's diagrams for perturbative quantum field theory, and Cvitanovic's birdtracks for Lie algebras. Category theory provides a robust framework in which to understand the nature of such diagrams, and Joyal and Street formalized this framework by introducing string diagrams, governed by the syntax of monoidal 1-categories. The notion of "manifold diagrams" generalizes string diagrams to higher dimensions, and can be interpreted in higher-categorical terms by a process of geometric dualization. The closely related notion of "tame tangles" describes a well-behaved class of embedded manifolds that can likewise be interpreted categorically. In this paper we formally introduce the notions of manifold diagrams and of tame tangles, and show that they admit a combinatorial classification, by using results from the toolbox of framed combinatorial topology. We then study the stability of tame tangles under perturbation; the local forms of perturbation stable tame tangles provide combinatorial models of differential singularities. As an illustration we describe various such combinatorial singularities in low dimensions. We conclude by observing that all smooth 4-manifolds can be presented as tame tangles, and conjecture that the same is true for smooth manifolds of any dimension.