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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
References
Publication
Received: 7 February 2021
Revised: 27 July 2021
Accepted: 15 August 2021
Published: 13 December 2022
Authors
Tejas Kalelkar
Mathematics Department
Indian Institute of Science Education and Research Pune
Pune
India
Sriram Raghunath
Mathematics Department
Indian Institute of Science Education and Research Pune
Pune
India