Bounds on Pachner moves and systoles of cusped 3-manifolds

Abstract

Any two geometric ideal triangulations of a cusped complete hyperbolic $3$-manifold $M$ are related by a sequence of Pachner moves through topological triangulations. We give a bound on the length of this sequence in terms of the total number of tetrahedra and a lower bound on dihedral angles. This leads to a naive but effective algorithm to check if two hyperbolic knots are equivalent, given geometric ideal triangulations of their complements. Given a geometric ideal triangulation of $M$, we also give a lower bound on the systole length of $M$ in terms of the number of tetrahedra and a lower bound on dihedral angles.Comment: Exactly the same arguments work for hyperbolic manifolds with multiple cusps, so statements of theorems are generalised from one-cusped hyperbolic manifolds to cusped hyperbolic manifold