Volume 9, issue 1 (2009)

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Bitwist 3–manifolds

J W Cannon, W J Floyd and W R Parry

Algebraic & Geometric Topology 9 (2009) 187–220

Our earlier twisted-face-pairing construction showed how to modify an arbitrary orientation-reversing face-pairing on a faceted 3–ball in a mechanical way so that the quotient is automatically a closed, orientable 3–manifold. The modifications were, in fact, parametrized by a finite set of positive integers, arbitrarily chosen, one integer for each edge class of the original face-pairing. This allowed us to find very simple face-pairing descriptions of many, though presumably not all, 3–manifolds.

Here we show how to modify the construction to allow negative parameters, as well as positive parameters, in the twisted-face-pairing construction. We call the modified construction the bitwist construction. We prove that all closed connected orientable 3–manifolds are bitwist manifolds. As with the twist construction, we analyze and describe the Heegaard splitting naturally associated with a bitwist description of a manifold.

3–manifold constructions, Dehn surgeries
Mathematical Subject Classification 2000
Primary: 57N10
Received: 13 June 2008
Revised: 13 January 2009
Accepted: 13 November 2008
Published: 2 February 2009
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