  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.
