Geodesic
Heegaard diagrams and surgery descriptions for twisted face-pairing 3-manifolds
Cannon, James W1 ; Floyd, W J2 ; Parry, W R3
1Department of Mathematics, Brigham Young University, Provo, UT 84602, USA
2Department of Mathematics, Virginia Tech, Blacksburg VA 24061, USA
3Department of Mathematics, Eastern Michigan University, Ypsilanti MI 48197, USA
Algebraic and Geometric Topology, Tome 3 (2003) no. 1, pp. 235-285
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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.

DOI : 10.2140/agt.2003.3.235
Keywords: 3–manifold constructions, Dehn surgery, Heegaard diagrams

Cannon, James W  1   ; Floyd, W J  2   ; Parry, W R  3

1 Department of Mathematics, Brigham Young University, Provo, UT 84602, USA
2 Department of Mathematics, Virginia Tech, Blacksburg VA 24061, USA
3 Department of Mathematics, Eastern Michigan University, Ypsilanti MI 48197, USA 
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Cannon, James W; Floyd, W J; Parry, W R. Heegaard diagrams and surgery descriptions for twisted face-pairing 3-manifolds. Algebraic and Geometric Topology, Tome 3 (2003) no. 1, pp. 235-285. doi: 10.2140/agt.2003.3.235

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[2] J W Cannon, W J Floyd, W R Parry, Twisted face-pairing 3–manifolds, Trans. Amer. Math. Soc. 354 (2002) 2369

[3] J W Cannon, W J Floyd, W R Parry, Ample twisted face-pairing 3–manifolds, preprint

[4] J W Cannon, W J Floyd, W R Parry, A survey of twisted face-pairing 3–manifolds, preprint

[5] J M Montesinos, Classical tessellations and three-manifolds, Universitext, Springer (1987)

[6] V V Prasolov, A B Sossinsky, Knots, links, braids and 3–manifolds, Translations of Mathematical Monographs 154, American Mathematical Society (1997)

[7] W P Thurston, The Geometry and Topology of Three-Manifolds, lecture notes, Princeton University (1980)

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