Bitwist 3-manifolds

TL;DR

This paper introduces the bitwist construction, an extension of previous face-pairing methods, enabling the description of all closed orientable 3-manifolds through a parametrized, flexible approach.

Contribution

It extends the twisted-face-pairing construction to include negative parameters, proving that all closed orientable 3-manifolds can be represented as bitwist manifolds.

Findings

All closed orientable 3-manifolds are bitwist manifolds.

The construction allows negative and positive parameters.

Heegaard splittings are naturally described within this framework.

Abstract

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.