== polytopes with at most 7 vertices: cyclic, stacked, pyramids, bipyramids
4-simplex                          f=(5, 10, 10, 5)   pairs=0      nonzero=0     odd=0     max|lk|=0 linked=False minor=None apex=5
ok     consistent with Corollary 1.2 (non-zero pair <=> odd pair <=> minor), second diagram agrees 
C(6,4)                             f=(6, 15, 18, 9)   pairs=10     nonzero=1     odd=1     max|lk|=1 linked=True minor=K6 apex=0
ok     consistent with Corollary 1.2 (non-zero pair <=> odd pair <=> minor), second diagram agrees 
C(7,4)                             f=(7, 21, 28, 14)  pairs=175    nonzero=21    odd=21    max|lk|=1 linked=True minor=K6 apex=0
ok     consistent with Corollary 1.2 (non-zero pair <=> odd pair <=> minor), second diagram agrees 
stacked, 6 vertices                f=(6, 14, 16, 8)   pairs=6      nonzero=0     odd=0     max|lk|=0 linked=False minor=None apex=4
ok     consistent with Corollary 1.2 (non-zero pair <=> odd pair <=> minor), second diagram agrees 
stacked, 7 vertices                f=(7, 18, 22, 11)  pairs=51     nonzero=0     odd=0     max|lk|=0 linked=False minor=None apex=3
ok     consistent with Corollary 1.2 (non-zero pair <=> odd pair <=> minor), second diagram agrees 
pyramid over square pyramid        f=(6, 13, 13, 6)   pairs=4      nonzero=0     odd=0     max|lk|=0 linked=False minor=None apex=6
ok     consistent with Corollary 1.2 (non-zero pair <=> odd pair <=> minor), second diagram agrees 
pyramid over triangular bipyramid  f=(6, 14, 15, 7)   pairs=6      nonzero=0     odd=0     max|lk|=0 linked=False minor=None apex=4
ok     consistent with Corollary 1.2 (non-zero pair <=> odd pair <=> minor), second diagram agrees 
bipyramid over tetrahedron         f=(6, 14, 16, 8)   pairs=6      nonzero=0     odd=0     max|lk|=0 linked=False minor=None apex=4
ok     consistent with Corollary 1.2 (non-zero pair <=> odd pair <=> minor), second diagram agrees 
pyramid over octahedron            f=(7, 18, 20, 9)   pairs=64     nonzero=0     odd=0     max|lk|=0 linked=False minor=None apex=1
ok     consistent with Corollary 1.2 (non-zero pair <=> odd pair <=> minor), second diagram agrees 
pyramid over triangular prism      f=(7, 15, 14, 6)   pairs=16     nonzero=0     odd=0     max|lk|=0 linked=False minor=None apex=7
ok     consistent with Corollary 1.2 (non-zero pair <=> odd pair <=> minor), second diagram agrees 
bipyramid over square pyramid      f=(7, 18, 21, 10)  pairs=64     nonzero=0     odd=0     max|lk|=0 linked=False minor=None apex=1
ok     consistent with Corollary 1.2 (non-zero pair <=> odd pair <=> minor), second diagram agrees 
bipyramid over triangular bipyramid f=(7, 19, 24, 12)  pairs=85     nonzero=1     odd=1     max|lk|=1 linked=True minor=K6 apex=0
ok     consistent with Corollary 1.2 (non-zero pair <=> odd pair <=> minor), second diagram agrees 
ok   pyramids: the apex has the property of Lemma 6.1; simplex: every vertex 
ok   report of the first run: 6 pairs for the bipyramid over the tetrahedron and for the pyramid over the triangular bipyramid, 64 for the pyramid over the octahedron, all zero; C(7,4): 175 pairs, 21 non-zero 
== prisms, products, and further named polytopes
prism over tetrahedron             f=(8, 16, 14, 6)   pairs=52     nonzero=0     odd=0     max|lk|=0 linked=False minor=None apex=8
ok     consistent with Corollary 1.2 (non-zero pair <=> odd pair <=> minor), second diagram agrees 
triangle x triangle                f=(9, 18, 15, 6)   pairs=78     nonzero=0     odd=0     max|lk|=0 linked=False minor=None apex=9
ok     consistent with Corollary 1.2 (non-zero pair <=> odd pair <=> minor), second diagram agrees 
cross-polytope                     f=(8, 24, 32, 16)  pairs=691    nonzero=3     odd=3     max|lk|=1 linked=True minor=K6 apex=0
ok     consistent with Corollary 1.2 (non-zero pair <=> odd pair <=> minor), second diagram agrees 
C(8,4)                             f=(8, 28, 40, 20)  pairs=2107   nonzero=305   odd=304   max|lk|=2 linked=True minor=K6 apex=0
ok     consistent with Corollary 1.2 (non-zero pair <=> odd pair <=> minor), second diagram agrees 
stacked, 8 vertices                f=(8, 22, 28, 14)  pairs=231    nonzero=0     odd=0     max|lk|=0 linked=False minor=None apex=3
ok     consistent with Corollary 1.2 (non-zero pair <=> odd pair <=> minor), second diagram agrees 
stacked, 9 vertices (another order) f=(9, 26, 34, 17)  pairs=717    nonzero=0     odd=0     max|lk|=0 linked=False minor=None apex=1
ok     consistent with Corollary 1.2 (non-zero pair <=> odd pair <=> minor), second diagram agrees 
triangle x square                  f=(12, 24, 19, 7)  partitions=526 basis pairs=1579 nonzero=0 linked=False minor=None apex=0
ok     consistent with Corollary 1.2 (non-zero pair <=> odd pair <=> minor), second diagram agrees 
prism over octahedron              f=(12, 30, 28, 10) partitions=220 basis pairs=1933 nonzero=1 linked=True minor=K6 apex=0
ok     consistent with Corollary 1.2 (non-zero pair <=> odd pair <=> minor), second diagram agrees 
triangle x pentagon                f=(15, 30, 23, 8)  partitions=5613 basis pairs=21888 nonzero=0 linked=False apex=0
ok     consistent with Corollary 1.2 (non-zero pair <=> odd pair <=> minor), second diagram agrees 
triangle x hexagon                 f=(18, 36, 27, 9)  partitions=12563 basis pairs=65051 nonzero=1 linked=True apex=0
ok     consistent with Corollary 1.2 (non-zero pair <=> odd pair <=> minor), second diagram agrees 
4-cube                             f=(16, 32, 24, 8)  partitions=2550 basis pairs=10704 nonzero=1 linked=True apex=0
ok     consistent with Corollary 1.2 (non-zero pair <=> odd pair <=> minor), second diagram agrees 
ok   prism over the tetrahedron: 52 pairs of disjoint cycles, product of two triangles: 78, all with linking number 0 (numbers of the first run) 
ok   cross-polytope: 691 pairs, 3 of them non-zero, with |lk| = 1 (numbers of the first run) 
ok   Eppstein: the 4-cube and the (3,6)-duoprism are linked; pyramids and stacked polytopes are not; (3,5)-duoprism not linked (second run) 
== Section 6.1: Lemma 6.1 and its examples
ok   prism over the tetrahedron: every vertex has the property of Lemma 6.1 
ok     the facets through a are the bottom (4 vertices) and three prisms over triangles (6 vertices) 
ok   product of two triangles: every vertex has the property of Lemma 6.1 
ok   product of a triangle and a square: no vertex has the property; no non-zero pair (complete search); no minor in the Petersen family (exhaustive) partitions examined for the six other graphs: 66005
== Section 6.2: C(6,4)
ok   the facets are the complements of the pairs {i,j} with j - i odd; there are nine 
ok   all triangles are 2-faces except 135 and 246 
ok   ten pairs of disjoint cycles; exactly one has non-zero linking number: {135, 246}, value +-1 
ok   the graph of C(6,4) is K_6 
== Section 6.3: the prism over the octahedron
ok   12 vertices, 30 edges: on each level adjacent unless antipodal, and every vertex joined to its copy 
ok   facets: two octahedra and eight prisms over triangles 
ok   the six sets of the note are a K_6 model (partition of the 12 vertices, connected, pairwise adjacent) 
ok   the fifteen witness edges of the note are edges and join the fifteen pairs of sets 
ok   C and D are disjoint 6-cycles with linking number +-1 (three Schlegel diagrams; crossings over, under, Gauss) [(1, 1, 1.0), (1, 1, 1.0), (-1, -1, -1.0)]
ok     the projection of C is the circle z = 0 (x+, y+, x-, y-), that of D the circle x = 0; over y+ the cycle C is on level 0 and D on level 1, over y- the other way round 
== Figure 1
ok   the direction (1, -2, 1.8) is the kernel of the projection of the figure 
ok   the twelve points of the TikZ picture are the projections of the twelve vertices 
     crossing at (-0.1250, 0.5750): C edge ('y+', 0)-('x-', 0), D edge ('z+', 0)-('y-', 0); C nearer to a viewer in direction +k: False; sign factor 1
     crossing at (0.7500, 0.1500): C edge ('x+', 0)-('y+', 0), D edge ('z-', 1)-('y+', 1); C nearer to a viewer in direction +k: True; sign factor -1
     crossing at (0.3000, -1.3800): C edge ('y-', 1)-('x+', 1), D edge ('z-', 1)-('y+', 1); C nearer to a viewer in direction +k: True; sign factor 1
     crossing at (0.0000, -1.4400): C edge ('y-', 1)-('x+', 1), D edge ('z-', 0)-('z-', 1); C nearer to a viewer in direction +k: True; sign factor -1
ok   the projections of C and D cross exactly four times, at the four places marked in the picture 
ok   the over/under information of the picture is that of one viewer (the same side for all four crossings) viewer in direction +(1, -2, 1.8)
ok   C lies over D at three of the four crossings; the signed sums over the crossings with C over D and with D over C are equal, of absolute value 1 (-1, -1)
     (a degenerate position of two edges of one cycle in the picture)
     (a degenerate position of two edges of one cycle in the picture)
     self-crossings of the projections of C and of D in the picture: 0
ok   in the model of two concentric octahedra (scales 1 and 2.4) the two polygons have linking number +-1 (-1, -1, -1.000000000)
== Section 6.4: the truncated 4-simplex, the hypersimplex and its polar
ok   truncated 4-simplex: f-vector (20, 40, 30, 10) 
ok     (i,j) is adjacent to (j,i) and to the three vertices (i,k) 
ok     it is the polar of the simplex stacked on all five facets (isomorphic face lattices) 
ok     C and D are disjoint 10-cycles through all 20 vertices 
ok     lk(C, D) = +-1 (three Schlegel diagrams, based at facets with 4 and 12 vertices; crossings over, under, Gauss) [(-1, -1, -1.0, 12), (-1, -1, -1.000000002, 4), (1, 1, 1.0, 4)]
ok     the six sets of the note are a K_6 model: a partition of the 20 vertices into connected, pairwise adjacent sets; the last two induce paths in the order written 
ok     the fifteen witness edges of the note are edges and join the fifteen pairs of sets 
hypersimplex Delta(2,5)            f=(10, 30, 30, 10) partitions=142 basis pairs=1257 nonzero=1 linked=True minor=K6 apex=0
ok     consistent with Corollary 1.2 (non-zero pair <=> odd pair <=> minor), second diagram agrees 
ok     Delta(2,5) is linked and has a K_6-minor (model found by exhaustive search and verified) model: [[0], [1], [2], [3, 4], [5, 6, 9], [7, 8]]
polar of Delta(2,5)                f=(10, 30, 30, 10) partitions=280 basis pairs=1920 nonzero=0 linked=False minor=None apex=0
ok     consistent with Corollary 1.2 (non-zero pair <=> odd pair <=> minor), second diagram agrees 
ok     the polar of Delta(2,5): no non-zero pair (complete search), no minor in the Petersen family (exhaustive), no vertex with the property of Lemma 6.1 partitions examined for the six other graphs: 5031
ok     its graph: five vertices of degree 4 and five of degree 8 (K_5 with a vertex attached to each of its five K_4) 
time 256.3 s
ALL OK
