LCG moves in crystallizations (Q1303805)

The paper deals with two combinatorial representations of closed 3-manifolds by means of crystallizations (a special kind of 4-colored 4-regular graph) and Heegaard diagrams. Except coloring of its edges, a crystallization can be viewed as an extended Heegaard diagram. So the authors investigate relations between Singer moves or geometric \(T\)-transformations in Heegaard diagrams and polyhedral cut and glue moves in crystallizations. In particular, the authors introduce linear cut and glue moves (LCG moves), a particular kind of crystallization moves which amount to the replacement of meridians in extended Heegaard diagrams. Then they exhibit various applications of LCG moves, expecially on the equivalence of certain Lins crystallizations [\textit{S. Lins}, Gems, computers and attractors for 3-manifolds, Ser. Knots Everything 5 (1995; Zbl 0868.57002)]. In particular, they show that two inequivalent Heegaard splittings of the Brieskorn homology sphere \(\Sigma(2,3,7)\) are equivalent under single stabilization of each Heegaard splitting by using LCG moves associated with the stable equivalence.