  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.
