On Matveev-Piergallini moves for branched spines

The Matveev-Piergallini (MP) moves on spines of 3-manifolds are well-known for their correspondence with the Pachner 2-3 moves in dual ideal triangulations. Benedetti and Petronio introduced combinatorial descriptions of closed 3-manifolds and combed 3-manifolds using branched spines and their equivalence relations, which involve MP moves with 16 distinct branching patterns. In this talk, I will demonstrate that these 16 MP moves on branched spines are derived from a primary MP move, pure sliding moves, and their inverses. Consequently, we obtain alternative generating sets for the equivalence relations on branched spines for closed 3-manifolds and combed 3-manifolds, which provide a useful framework for studying quantum invariants using ideal triangulations. This is joint work with K. Muramatsu and K. Taguchi.