Compact 3-manifolds via 4-colored graphs

TL;DR

This paper presents a new graph-based method for representing compact 3-manifolds, facilitating computer-aided analysis, classification, and enumeration of these manifolds with small graph representations.

Contribution

It generalizes previous graph representations to compact 3-manifolds without spherical boundary components, enabling effective computational study and classification.

Findings

Established relations between graph moves and manifold equivalences

Classified manifolds with small graph representations

Analyzed fundamental groups and Heegaard genus in this context

Abstract

We introduce a representation of compact 3-manifolds without spherical boundary components via (regular) 4-colored graphs, which turns out to be very convenient for computer aided study and tabulation. Our construction is a direct generalization of the one given in the eighties by S. Lins for closed 3-manifolds, which is in turn dual to the earlier construction introduced by Pezzana's school in Modena. In this context we establish some results concerning fundamental groups, connected sums, moves between graphs representing the same manifold, Heegaard genus and complexity, as well as an enumeration and classification of compact 3-manifolds representable by graphs with few vertices ($\le 6$ in the non-orientable case and $\le 8$ in the orientable one).