# HG changeset patch
# User Frederic Chapoton
# Date 1356871928 -3600
# Node ID 15bb8b362364e04fb0ce9a754cfec237f458a045
# Parent fe34fbbd2093f6371cbc1c2e1aa2bb5674ddb43b
#13888 Defines the Barnette triangulation of the 3-sphere
* * *
Barnette Sphere - reviewer's patch
diff --git a/sage/homology/examples.py b/sage/homology/examples.py
--- a/sage/homology/examples.py
+++ b/sage/homology/examples.py
@@ -27,6 +27,7 @@ All of these examples are accessible by
You can get a list by typing ``simplicial_complexes.`` and hitting the
TAB key::
+ simplicial_complexes.BarnetteSphere
simplicial_complexes.ChessboardComplex
simplicial_complexes.ComplexProjectivePlane
simplicial_complexes.K3Surface
@@ -158,6 +159,7 @@ class SimplicialComplexExamples():
Here are the available examples; you can also type
``simplicial_complexes.`` and hit tab to get a list:
+ - :meth:`BarnetteSphere`
- :meth:`ChessboardComplex`
- :meth:`ComplexProjectivePlane`
- :meth:`K3Surface`
@@ -815,6 +817,59 @@ class SimplicialComplexExamples():
(1, 2, 4, 7, 15), (2, 3, 7, 8, 16), (1, 4, 5, 6, 10)],
is_mutable=False)
+ def BarnetteSphere(self):
+ r"""
+ Returns Barnette's triangulation of the 3-sphere.
+
+ This is a pure simplicial complex of dimension 3 with 8
+ vertices and 19 facets, which is a non-polytopal triangulation
+ of the 3-sphere. It was constructed by Barnette in
+ [B1970]_. The construction here uses the labeling from De
+ Loera, Rambau and Santos [DLRS2010]_. Another reference is chapter
+ III.4 of Ewald [E1996]_.
+
+ EXAMPLES::
+
+ sage: BS = simplicial_complexes.BarnetteSphere() ; BS
+ Simplicial complex with vertex set (1, 2, 3, 4, 5, 6, 7, 8) and 19 facets
+ sage: BS.f_vector()
+ [1, 8, 27, 38, 19]
+
+ TESTS:
+
+ Checks that this is indeed the same Barnette Sphere as the one
+ given on page 87 of [E1996]_.::
+
+ sage: BS2 = SimplicialComplex([[1,2,3,4],[3,4,5,6],[1,2,5,6],
+ ... [1,2,4,7],[1,3,4,7],[3,4,6,7],
+ ... [3,5,6,7],[1,2,5,7],[2,5,6,7],
+ ... [2,4,6,7],[1,2,3,8],[2,3,4,8],
+ ... [3,4,5,8],[4,5,6,8],[1,2,6,8],
+ ... [1,5,6,8],[1,3,5,8],[2,4,6,8],
+ ... [1,3,5,7]])
+ sage: BS.is_isomorphic(BS2)
+ True
+
+ REFERENCES:
+
+ .. [B1970] Barnette, "Diagrams and Schlegel diagrams", in
+ Combinatorial Structures and Their Applications, Proc. Calgary
+ Internat. Conference 1969, New York, 1970, Gordon and Breach.
+
+ .. [DLRS2010] De Loera, Rambau and Santos, "Triangulations:
+ Structures for Algorithms and Applications", Algorithms and
+ Computation in Mathematics, Volume 25, Springer, 2011.
+
+ .. [E1996] Ewald, "Combinatorial Convexity and Algebraic Geometry",
+ vol. 168 of Graduate Texts in Mathematics, Springer, 1996
+
+ """
+ return SimplicialComplex([
+ (1,2,4,5),(2,3,5,6),(1,3,4,6),(1,2,3,7),(4,5,6,7),(1,2,4,7),
+ (2,4,5,7),(2,3,5,7),(3,5,6,7),(3,1,6,7),(1,6,4,7),(1,2,3,8),
+ (4,5,6,8),(1,2,5,8),(1,4,5,8),(2,3,6,8),(2,5,6,8),(3,1,4,8),
+ (3,6,4,8)])
+
###############################################################
# examples from graph theory: