Data for SnapPy manifolds

• Trace fields (of geometric representation)
  • OrientableCuspedCensus (explanation)
    • up to 8 tetrahedra
    • 9 tetrahedra
• PGL(2,C)-representations
  • OrientableCuspedCensus (explanation)

    2 tetrahedra 3 tetrahedra 4 tetrahedra 5 tetrahedra

    6 tetrahedra 7 tetrahedra 8 tetrahedra 9 tetrahedra

  • LinkExteriors (explanation)

    2 tetrahedra 3 tetrahedra 4 tetrahedra 5 tetrahedra 6 tetrahedra 7 tetrahedra 8 tetrahedra 9 tetrahedra

    10 tetrahedra 11 tetrahedra 12 tetrahedra 13 tetrahedra 14 tetrahedra 15 tetrahedra 16 tetrahedra 17 tetrahedra: all but 11_2, 11_148, 11_357, 11_367, 11_369

  • HTLinkExteriors (explanation)

    2 tetrahedra 3 tetrahedra 4 tetrahedra 5 tetrahedra 6 tetrahedra 7 tetrahedra

    8 tetrahedra 9 tetrahedra 10 tetrahedra 11 tetrahedra 12 tetrahedra 13 tetrahedra: all but L14n31302

• PGL(3,C)-representations
  • OrientableCuspedCensus (explanation)
    • 2 tetrahedra
    • 3 tetrahedra
    • 4 tetrahedra: m022, m023, m026, m027, m029, m030, m032, m033, m034, m035, m036, m037, m038, m043, m044, m045, m046, m052, m053, m054, m055, m060, m069, m070, m081, m082, m100, m117, m125, m129, m130, m135, m136, m137, m142, m146, m148, m149, m202, m203
• PGL(4,C)-representations
  • OrientableCuspedCensus (explanation)
    • 2 tetrahedra: m004

Explanations (also see notes and caveats)

• Manifold: Name of manifold in SnapPy census. We use the SnapPy triangulation.
• Obstruction index: Representations are grouped according to their obstruction class to lifting the representation to a boundary-unipotent SL(N,C)-representation. An obstruction class is an element in H²(M, bd M; Z/N) modulo the action by units. The trivial class always has index 0. See Manifold.ptolemy_generalized_obstruction_classes.
• Degree of component: Each representation lies in a component of the Ptolemy variety, which is either zero-dimensional or not. Representations in a zero-dimensional component are grouped together according to Galois conjugacy, the degree being the number of Galois conjugates. The degree is the degree of the shape field, not the Ptolemy field. Each 0-dimensional representation appears at most once in the table.
• SL(2,C): A checkmark indicates that a representation is induced from a representation in SL(2,C) via the canonical irreducible map SL(2,C)->SL(N,C).
  See from snappy.ptolemy.coordinates import CrossRatios; help(CrossRatios.induced_representation).
• PSL(2,C): A checkmark indicates that a representation is induced from a representation in PSL(2,C) via the canonical irreducible map PSL(2,C)->pSL(N,C) = SL(N,C) / {-1^N+1 I}.
  See from snappy.ptolemy.coordinates import CrossRatios; help(CrossRatios.is_induced_from_psl2).
• PSL(3,R): A checkmark indicates that a representation is real.
  See from snappy.ptolemy.coordinates import CrossRatios; help(CrossRatios.is_real).
• SL(3,C): A checkmark indicates that the PGL(3,C)-representation lifts to a SL(3,C)-representation.
  This is true if and only if the obstruction class is trivial, i.e., has index 0. We do not distinguish between different lifts to SL(3,C).
• PU(2,1): A checkmark indicates that the representation has image in the subgroup PU(2,1).
  We use the conditions in Falbel, Koseleff, Rouillier: Representations of Fundamental Groups of 3-Manifolds into PGL(3,C): Exact Computations in Low Complexity.
  See from snappy.ptolemy.coordinates import CrossRatios; help(CrossRatios.is_cr_structure).
• Complex volume: Cheeger-Chern-Simons invariant of a representation divided by i as defined in Garoufalidis, Thurston, Zickert: The complex volume of SL(n,C)-representations of 3-manifolds. For the geometric representation of a hyperbolic manifold, this is equal to volume + i Chern-Simons and marked in bold. For pSL(N,C), we compute it modulo i pi²/6. For PGL(3,C), we compute it modulo i pi²/18.
  See from snappy.ptolemy.coordinates import PtolemyCoordinates; help(PtolemyCoordinates.complex_volume_numerical).