4-colored graphs and knot/link complements

TL;DR

This paper classifies orientable 3-manifolds with toric boundary, including knot and link complements, using 4-colored graphs up to 12 vertices, and extends the classification for knot complements up to 16 vertices.

Contribution

It provides a complete catalog of such 3-manifolds up to 12 vertices and extends the classification of knot complements up to 16 vertices, advancing the understanding of their combinatorial representations.

Findings

Complete classification of these manifolds up to 12 vertices.

Diagrams of knots and links involved are provided.

Extended classification for knot complements up to 16 vertices.

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.