The twisted face-pairing construction of our earlier papers gives an efficient way of
generating, mechanically and with little effort, myriads of relatively simple
face-pairing descriptions of interesting closed 3–manifolds. The corresponding
description in terms of surgery, or Dehn-filling, reveals the twist construction as a
carefully organized surgery on a link. In this paper, we work out the relationship
between the twisted face-pairing description of closed 3–manifolds and the more
common descriptions by surgery and Heegaard diagrams. We show that all
Heegaard diagrams have a natural decomposition into subdiagrams called
Heegaard cylinders, each of which has a natural shape given by the ratio
of two positive integers. We characterize the Heegaard diagrams arising
naturally from a twisted face-pairing description as those whose Heegaard
cylinders all have integral shape. This characterization allows us to use the
Kirby calculus and standard tools of Heegaard theory to attack the problem
of finding which closed, orientable 3–manifolds have a twisted face-pairing
description.
