Face pairing graphs and 3-manifold enumeration

Abstract

The face pairing graph of a 3-manifold triangulation is a 4-valent graph denoting which tetrahedron faces are identified with which others. We present a series of properties that must be satisfied by the face pairing graph of a closed minimal P^2-irreducible triangulation. In addition we present constraints upon the combinatorial structure of such a triangulation that can be deduced from its face pairing graph. These results are then applied to the enumeration of closed minimal P^2-irreducible 3-manifold triangulations, leading to a significant improvement in the performance of the enumeration algorithm. Results are offered for both orientable and non-orientable triangulations.

30 pages, 57 figures; v2: clarified some passages and generalised the final theorem to the non-orientable case; v3: fixed a flaw in the proof of the conical face lemma

Cited by in corpus (8)

• The Pachner graph and the simplification of 3-sphere triangulations
• Structures of small closed non-orientable 3-manifold triangulations
• A new approach to crushing 3-manifold triangulations
• Detecting genus in vertex links for the fast enumeration of 3-manifold triangulations
• Maximal admissible faces and asymptotic bounds for the normal surface solution space
• Bounds for the genus of a normal surface
• Fixed parameter tractable algorithms in combinatorial topology
• Knotted 4-regular graphs: polynomial invariants and the Pachner moves