Bounds on Pachner moves and systoles of cusped 3-manifolds

TL;DR

This paper establishes bounds on the number of Pachner moves needed to relate different ideal triangulations of cusped hyperbolic 3-manifolds and provides bounds on systole lengths, aiding in knot equivalence checks.

Contribution

It introduces explicit bounds on Pachner move sequences and systole lengths based on tetrahedra count and dihedral angles, advancing understanding of hyperbolic 3-manifold triangulations.

Findings

Bound on Pachner move sequence length in terms of tetrahedra and angles.

Lower bound on systole length based on triangulation data.

Algorithm for knot equivalence verification using geometric triangulations.

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.