Face Pairing Graphs and 3-manifold Enumeration
نویسندگان
چکیده
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-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-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.
منابع مشابه
Enumeration of non-orientable 3-manifolds using face pairing graphs and union-find
Drawing together techniques from combinatorics and computer science, we improve the census algorithm for enumerating closed minimal P-irreducible 3-manifold triangulations. In particular, new constraints are proven for face pairing graphs, and pruning techniques are improved using a modification of the union-find algorithm. Using these results we catalogue all 136 closed nonorientable P-irreduc...
متن کاملFixed Parameter Tractable Algorithms in Combinatorial Topology
To enumerate 3-manifold triangulations with a given property, one typically begins with a set of potential face pairing graphs (also known as dual 1-skeletons), and then attempts to flesh each graph out into full triangulations using an exponential-time enumeration. However, asymptotically most graphs do not result in any 3-manifold triangulation, which leads to significant “wasted time” in top...
متن کاملTwisted Face-pairing 3-manifolds
This paper is an enriched version of our introductory paper on twisted face-pairing 3-manifolds. Just as every edge-pairing of a 2-dimensional disk yields a closed 2-manifold, so also every face-pairing of a faceted 3ball P yields a closed 3-dimensional pseudomanifold. In dimension 3, the pseudomanifold may suffer from the defect that it fails to be a true 3-manifold at some of its vertices. Th...
متن کاملHeegaard Diagrams and Surgery Descriptions for Twisted Face-pairing 3-manifolds
We give a simple algorithmic construction of a Heegaard diagram for an arbitrary twisted face-pairing 3-manifold. One family of meridian curves in the Heegaard diagram corresponds to the face pairs, and the other family is obtained from the first by a product of powers of Dehn twists. These Dehn twists are along curves which correspond to the edge cycles and the powers are the multipliers. From...
متن کاملA CHARACTERIZATION FOR METRIC TWO-DIMENSIONAL GRAPHS AND THEIR ENUMERATION
The textit{metric dimension} of a connected graph $G$ is the minimum number of vertices in a subset $B$ of $G$ such that all other vertices are uniquely determined by their distances to the vertices in $B$. In this case, $B$ is called a textit{metric basis} for $G$. The textit{basic distance} of a metric two dimensional graph $G$ is the distance between the elements of $B$. Givi...
متن کامل