Abstract. In [11] we parametrized boundary-unipotent representations of a 3-manifold group into SL(n, C) using Ptolemy coordinates, which were inspired by A-coordinates on higher Teichmüller space due to Fock and Goncharov. In this paper, we parametrize representations into PGL(n, C) using shape coordinates which are a 3-dimensional analogue of Fock and Goncharov's X -coordinates. These coordinates satisfy equations generalizing Thurston's gluing equations. These equations are of Neumann-Zagier type and satisfy symplectic relations with applications in quantum topology. We also explore a duality between the Ptolemy coordinates and the shape coordinates.

Abstract. The Ptolemy coordinates for boundary-unipotent SL(n, C)-representations of a 3-manifold group were introduced in [7] inspired by the A-coordinates on higher Teichmüller space due to Fock and Goncharov. In this paper, we define the Ptolemy field of a (generic) PSL(2, C)-representation and prove that it coincides with the trace field of the representation. This gives an efficient algorithm to compute the trace field of a cusped hyperbolic manifold.

We call a cusped hyperbolic 3-manifold tetrahedral if it can be decomposed into regular ideal tetrahedra. Following an earlier publication by three of the authors, we give a census of all tetrahedral manifolds and all of their combinatorial tetrahedral tessellations with at most 25 (orientable case) and 21 (non-orientable case) tetrahedra. Our isometry classification uses certified canonical cell decompositions (based on work by Dunfield, Hoffman, Licata) and isomorphism signatures (an improvement of dehydration sequences by Burton). The tetrahedral census comes in Regina as well as SnapPy format, and we illustrate its features.

Abstract. The Ptolemy variety for SL(2, C) is an invariant of a topological ideal triangulation of a compact 3-manifold M . It is closely related to Thurston's gluing equation variety. The Ptolemy variety maps naturally to the set of conjugacy classes of boundary-unipotent SL(2, C)-representations, but (like the gluing equation variety) it depends on the triangulation, and may miss several components of representations. In this paper, we define a Ptolemy variety, which is independent of the choice of triangulation, and detects all boundary-unipotent irreducible SL(2, C)-representations. We also define variants of the Ptolemy variety for PSL(2, C)-representations, and representations that are not necessarily boundary-unipotent. In particular, we obtain an algorithm to compute all irreducible SL(2, C)-characters as well as the full A-polynomial. All the varieties are topological invariants of M .

This report lists the link diagrams in S 3 for all principal congruence link complements for which such a link diagram is known. Several unpublished link diagrams are included. Related to this, we also include one link diagram for an arithmetic regular tessellation link complement.