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Heegaard diagrams and surgery descriptions for twisted face-pairing 3-manifolds

J W Cannon, W J Floyd and W R Parry

Algebraic & Geometric Topology 3 (2003) 235–285

arXiv: math.GT/0303081


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.

3–manifold constructions, Dehn surgery, Heegaard diagrams
Mathematical Subject Classification 2000
Primary: 57N10
Forward citations
Received: 12 November 2001
Revised: 5 February 2003
Accepted: 14 February 2003
Published: 5 March 2003
J W Cannon
Department of Mathematics
Brigham Young University
Provo, UT 84602
W J Floyd
Department of Mathematics
Virginia Tech
Blacksburg VA 24061
W R Parry
Department of Mathematics
Eastern Michigan University
Ypsilanti MI 48197