Abstract
A representation for compact 3-manifolds with non-empty non-spherical boundary via 4-colored graphs (i.e., 4-regular graphs endowed with a proper edge-coloration with four colors) has been recently introduced by two of the authors, and an initial classification of such manifolds has been obtained up to 8 vertices of the representing graphs. Computer experiments show that the number of graphs/manifolds grows very quickly as the number of vertices increases. As a consequence, we have focused on the case of orientable 3-manifolds with toric boundary, which contains the important case of complements of knots and links in the 3-sphere. In this paper we obtain the complete catalogation/classification of these 3-manifolds up to 12 vertices of the associated graphs, showing the diagrams of the involved knots and links. For the particular case of complements of knots, the research has been extended up to 16 vertices.
Highlights
Introduction and preliminariesThe representation of closed 3-manifolds by 4-colored graphs has been independently introduced in the seventies by M
A 4-colored graph is a regular graph of degree four, endowed with a proper edge-coloration with four colors, and it represents a closed 3-manifold if and only if a certain combinatorial condition is satisfied
The extension of the representation by 4-colored graphs to 3-manifolds with boundary has been performed in [8], where any 4-colored graph is associated to a compact 3-manifold with boundary without spherical components, this correspondence being surjective on the whole class of such manifolds
Summary
The representation of closed 3-manifolds by 4-colored graphs has been independently introduced in the seventies by M. (a) the census of all non-isomorphic contracted bipartite 4-colored graphs (without 2dipoles) up to some order, representing compact orientable 3-manifolds with (possibly disconnected) toric boundary;. For (a), we wrote a script that, for each positive integer p, enumerates all non-isomorphic contracted 4-colored graphs with 2p vertices and no 2-dipoles, representing compact bordered 3-manifolds, possibly fixing the topological type of their boundary. For each 4-colored graph Γ this program constructed a triangulation of the manifold MΓ associated to Γ and obtained a census of compact orientable 3-manifolds with toric boundary. When more than one link was available a prime one was preferred (identified by its name in the Thistlethwaite link table)
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