diff --git a/doc_src/verify.rst b/doc_src/verify.rst index 642c5dae..051cc220 100644 --- a/doc_src/verify.rst +++ b/doc_src/verify.rst @@ -45,10 +45,9 @@ following computations: (Note that when using verified computation, the Chern-Simons invariant is only computed modulo pi^2/2 even though it is defined modulo pi^2.) -* Give the :doc:`canonical retriangulation ` (a close - relative to the canonical cell decomposition) of a cusped hyperbolic - manifold using intervals or exact arithmetic if necessary with - :py:meth:`~snappy.Manifold.canonical_retriangulation`:: +* Give the (a close relative to the canonical cell decomposition) of a cusped + hyperbolic manifold using intervals or exact arithmetic if necessary with + :meth:`~snappy.Manifold.canonical_retriangulation`:: sage: M = Manifold("m412") sage: K = M.canonical_retriangulation(verified = True) @@ -65,7 +64,7 @@ following computations: a higher value for :py:attr:`exact_bits_prec_and_degrees`. -* The :doc:`isometry signature ` which is a complete invariant of the isometry type +* The isometry signature which is a complete invariant of the isometry type of a cusped hyperbolic manifold (i.e., two manifolds are isometric if and only if they have the same isometry signature):: @@ -146,5 +145,4 @@ Verified computation topics .. toctree:: :maxdepth: 1 - verify_canon verify_internals diff --git a/doc_src/verify_canon.rst b/doc_src/verify_canon.rst deleted file mode 100644 index 8d8f05e5..00000000 --- a/doc_src/verify_canon.rst +++ /dev/null @@ -1,90 +0,0 @@ -Canonical retriangulation and isometry signature --------------------------------------------------------- - -The canonical retriangulation is a close relative to the canonical cell -decomposition defined by `Epstein and Penner -`_. -Like the canonical cell decomposition, it is intrinsic to -a hyperbolic manifold M and is (up to combinatorial isomorphism -relabeling the tetrahedra and vertices) completely determined by the -isometry type of a hyperbolic manifold. Unlike the canonical cell decomposition, -the canonical retriangulation always consists entirely of tetrahedra which makes -it more amenable for many computations by SnapPy. - -If the canonical cell decomposition of manifold M has only tetrahedral cells, -we define the canonical retriangulation to be the canonical cell decomposition. -In this case, the canonical retriangulation consists of ideal hyperbolic -tetrahedra and the ``canonical_retriangulation`` method returns a -SnapPy manifold. Example:: - - sage: M = Manifold("m015") - sage: K = M.canonical_retriangulation(verified = True) - sage: K.has_finite_vertices() # False iff all canonical cells tetrahedral - False - -If the canonical cell decomposition has non-tetrahedral cells, we turn it into -a topological triangulation as follows: pick a point (called center) in each -3-cell. "Suspend" each 2-cell (which is an ideal n-gon) between -the centers of the two neighboring 3-cells. These suspensions form a -decomposition of M into topological "diamonds". Each diamond can be split along -its central axis into n tetrahedra. This introduces finite vertices, thus -the ``verified_canonical_retriangulation`` method returns only a SnapPy -triangulation. Example (canonical cell is a cube):: - - sage: M = Manifold("m412") - sage: K = M.canonical_retriangulation(verified = True) - sage: K.has_finite_vertices() - True - -The canonical retriangulation can be used to certifiably find all isometries -of a manifold:: - - sage: K.isomorphisms_to(K) - [0 -> 1 1 -> 0 - [1 0] [1 0] - [0 1] [0 1] - Extends to link, - ... - Extends to link] - sage: len(K.isomorphisms_to(K)) - 8 - -Recall that the *isomorphism -signature* is a complete invariant of the combinatorial -isomorphism type of a triangulation that was defined by `Burton -`_. We can compute the isomorphism signature -of the canonical retriangulation:: - - sage: Manifold("m003").canonical_retriangulation(verified = True).triangulation_isosig() - 'cPcbbbdxm' - -The resulting invariant was called *isometry signature* by -`Goerner `_ and, for convenience, can be -accessed by:: - - sage: Manifold("m003").isometry_signature(verified = True) - 'cPcbbbdxm' - -It is a complete invariant of the isometry type of a hyperbolic manifold. -Thus it can be used to easily identify isometric manifolds -(here, the last two manifolds have the same isometry signature and thus -have to be isomorphic):: - - sage: Manifold("m003").isometry_signature(verified = True) - 'cPcbbbdxm' - sage: Manifold("m004").isometry_signature(verified = True) - 'cPcbbbiht' - sage: Manifold("4_1").isometry_signature(verified = True) - 'cPcbbbiht' - sage: Manifold("m004").isometry_signature(verified = True) == Manifold("4_1").isometry_signature(verified = True) - True - - -Other applications of the canonical retriangulation include the detection of -2-bridge knots. - -======================================= -Verifying the canonical retriangulation -======================================= - -.. autofunction:: snappy.verify.verified_canonical_retriangulation