Moves for standard skeleta of 3-manifolds with marked boundary

TL;DR

This paper investigates the moves needed to relate different standard skeleta of 3-manifolds with marked boundary, introducing new moves and conditions to determine sufficiency of these moves.

Contribution

It extends classical move sets for 3-manifold skeleta by introducing new moves and conditions, providing a comprehensive framework for relating skeleta.

Findings

Classical MP-move and V-move are insufficient for all cases.

New moves (CR, T1, T2) are introduced to complete the move set.

Conditions are identified under which move sets are sufficient.

Abstract

We prove that the classical set of moves for standard spines of 3-manifolds (i.e. the MP-move and the V-move) does not suffice to relate to each other any two standard skeleta of a 3-manifold with marked boundary. We also describe a condition on the 3-manifold with marked boundary that tells whether the generalised set of moves, made up of the MP-move and the L-move, suffices to relate to each other any two standard skeleta of the 3-manifold with marked boundary. For the 3-manifolds with marked boundary that do not fulfil this condition, we give three other moves: the CR-move, the T1-move and the T2-move. The first one is local and, with the MP-move and the L-move, suffices to relate to each other any two standard skeleta of a 3-manifold with marked boundary fulfilling another condition. For the universal case, we prove that the non-local T1-move and T2-move, with the MP-move and the L-move, suffice to relate to each other any two standard skeleta of a generic 3-manifold with marked boundary. As a corollary, we get that disc-replacements suffice to relate to each other any two standard skeleta of a 3-manifold with marked boundary.