4-colored graphs and complements of knots and links

A representation for compact 3-manifolds with non-empty non-spherical boundary via 4-colored graphs (i.e. regular 4-valent graphs endowed by a proper edge-coloration with four colors) has been introduced in [1], where an initial tabulation/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 rapidly with the increasing of the vertices. As a consequence we focused our attentions on the case of 3-manifolds which are the complements of knots or links in the 3-sphere. In this context we obtained the classification of these 3-manifolds, up to 12 vertices of the representing graphs, showing the type of the links involved (they are exactly 22).
For the particular case of knot complements, the classification has been recently extended up to 16 vertices: there are exactly two complements of knots in the 3-sphere, the trivial knot (6 vertices) and the trefoil knot (16 vertices).
All these results are contained in [2], which will soon appear on the arXiv.
Joint work with P. Cristofori, E. Fominykh and V. Tarkaev.
References:

• P. Cristofori, M. Mulazzani, "Compact $3$-manifolds via $4$-colored graphs", RACSAM, published online: 24 July 2015. arXiv:1304.5070.
• P. Cristofori, E. Fominykh, M. Mulazzani, V. Tarkaev, $4$-colored graphs and knot/link complements, Preprint, 2016.