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Bounds on Pachner moves and systoles of cusped $3$–manifolds

### Tejas Kalelkar and Sriram Raghunath

Algebraic & Geometric Topology 22 (2022) 2951–2996
DOI: 10.2140/agt.2022.22.2951
##### 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.

##### Keywords
Hauptvermutung, ideal triangulations, hyperbolic knots, Pachner moves, systole length
##### Mathematical Subject Classification
Primary: 57K10, 57K32, 57Q25