A duplicate pair in the SnapPea census

We identify a duplicate pair in the well-known Callahan-Hildebrand-Weeks census of cusped finite-volume hyperbolic 3-manifolds. Specifically, the six-tetrahedron non-orientable manifolds x101 and x103 are homeomorphic.Comment: 5 pages, 3 figures; v2: minor edits. To appear in Experimental Mathematic

On spherical CR uniformization of 3-manifolds

International audienceWe consider the three discrete representations in the Falbel-Koseleff-Rouillier census where the peripheral subgroups have cyclic holonomy. We show that two of these representations have conjugate images, even though they represent different 3-manifold groups. This illustrates the fact that a discrete representation $\pi_1(M)\rightarrow PU(2,1)$ with cyclic unipotent boundary holonomy is not in general the holonomy of a spherical CR uniformization of $M$

Bounds on Pachner moves and systoles of cusped 3-manifolds

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