Notes and caveats (back to ptolemy or database)
PGL(n,ℂ)-representations, obstruction classes and Ptolemy varieties.
Each manifold in the SnapPy census is regarded as a manifold M with boundary a union of c tori, together with a topological ideal triangulation. For each boundary-unipotent representation ρ of π1(M) in PGL(n,ℂ) there is an associated class in H2(M,∂M; Z/nZ), namely the obstruction class to lifting ρ to a boundary-unipotent SL(n,ℂ)-representation. Hence, the boundary-unipotent representations are naturally grouped according to their obstruction class.
For each class σ in H2(M,∂M; Z/nZ), the Ptolemy variety for σ is a variety Pσ(M) together with a map to the set of representations whose obstruction class is σ. There is an action of (ℂ*)c(n-1) on the Ptolemy variety, and the map to the set of representations is invariant under this action. The variety computed is the one obtained by fixing this action. Whenever referring to the Ptolemy variety, we always mean the one with the action fixed. Since the action on H2(M,∂M; Z/nZ) by units in Z/nZ changes the Ptolemy variety by a canonical isomorphism, we only consider obstruction classes up to the action by units.
Ptolemy coordinates and shapes.
The relationship between the Ptolemy variety and the gluing equation variety is summarized in Theorem 1.3 in "Gluing equations for PGL(n,C)-representations of 3-manifolds" together with the similar result (unpublished) involving general obstruction classes.
The set of all Ptolemy varieties maps surjectively to the gluing equation variety with finite fibers.
For the zero-dimensional components, we numerically compute for each Ptolemy assignment the corresponding shape assignment.
Hence, only the ones corresponding to distinct shape assignments are listed. For zero-dimensional components, distinct shape assignments correspond to distinct representations.
Why not use the gluing equations?
The Ptolemy varieties are much better suited for exact computations (using the Magma implementation of the F4 algorithm). When the number of simplices is less than 12, over 95% of the computations for n=2 finish in less than a second. The gluing equation varieties require significantly more time and memory, and often don't finish in a reasonable time even for a small number of simplices.
The reason for this significant difference is unclear, but may be because the Ptolemy equations are always homogeneous polynomials involving 3 terms of degree 2, whereas the gluing equations can have arbitrarily high degree. Also, for the gluing equations, substitutions of the form z'=1/(1-z) and z''=1-1/z introduce polynomials where the number of terms grows significantly with the degree of the edge.
Exact versus numerical
All computations of Ptolemy varieties are exact and are computed using the PrimaryDecomposition command in Magma.
The shapes are then computed numerically to check if two Ptolemy assignments correspond to the same representation.
It is well known that numerical computations by backwards solving from a Gröbner basis are very unstable, and small errors in the initial variable can lead to large errors in the final variable. We start by backwards solving the Ptolemy variety from the Gröbner basis with 100 digit precision. We then repeat increasing the precision until the gluing equations are satisfied with 50 digit precision. The gluing equations are thus guaranteed to hold with 50 digit precision. These computations are done in Pari.
The volumes are listed with 10 digit precision, but arbitrarily high precision can be obtained whenever needed.
When computing a Gröbner basis, the computation time and memory use is very sensitive to the choice of term order and other human input. We stress that all computations are done using the varieties given in SnapPy and that there is no preprocessing done to simplify the computations. All our computations can be independently verified without the need to modify the input to ease the computations.
Dependence of the triangulation
The Ptolemy variety depends on the triangulation of M, and is only computed for the census triangulations. The map from the Ptolemy variety to the set of representations is not surjective (for example, the reducible representations are never detected), and a different triangulation may detect different representations.
There is thus no guarantee that the set of volumes given is complete.
The same problem holds for the closely related gluing equation variety, and there is (to the best of our knowledge) no practical way around it.
It is proved in "The complex volume of SL(n,C)-representations of 3-manifolds" that by performing a single barycentric subdivision, one obtains a triangulation detecting all representations (even the reducible ones). This, however, is not practical, since the presence of interior vertices blows up the dimension of the varieties.
Higher dimensional components
For higher dimensional components, it is not necessarily true that distinct shape assignments correspond to distinct representations. For example, if there is a single cusp, a boundary-unipotent PGL(n,ℂ)-representation taking both meridian and longitude to the identity, will have a preimage in the gluing equation variety with dimension n(n-1)/2.
In general, the dimension of the preimage will depend on the image of the peripheral groups.
We do not currently check whether a higher dimensional component corresponds to a higher dimensional component of representations, or just a degeneration of the image of the peripheral groups.
The volume is constant on components, but we do not currently list the volume for the higher dimensional components.