J. W. Cannon, W. J. Floyd, and W. R. Parry
June 10, 2008
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.
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