This book is about dealing with 3-manifolds using computers. Its emphasis is on presenting algorithms which are used for solving (in practice) the homeomorphism problem for the smallest of these objects. The key concept is the 3-gem, a special kind of edge-colored graph, which encodes the manifold via a ball complex. Passages between 3-gems and more standard presentations like Heegaard diagrams and surgery descriptions are provided. A catalogue of all closed orientable 3-manifolds induced by 3-gems up to 30 vertices is included. In order to help the classification, various invariants are presented, including the new quantum invariants.
Part 1 Basic theory: E-graphs - special cell-decomposition of 2-manifolds; crystallizations - the Ferri-Gagliardi moves; diagrams of E-graphs and Ferri's switching lemma. Part 2 Generating surgery moves: quadricolours, hinges, commuters; relations among the generating moves; connections with Lickorish's construction. Part 3 Invariants: the fundamental and the homology groups; the vertex group ... is there something new here?; linking invariants. Part 4 Classes of 3-gems: the "planar" class and lens spaces; gists - special symmetries on 3-manifolds. Part 5 Theory for a catalogue of 3-gems: shortcuts for inserting the 4th colour; the TS-moves and the U-move. Appendices: all 3-gems up to 28 vertices. (Part Contents).