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Transcript
How is a graph like a manifold?
Ethan Bolker
Mathematics - UMass Boston
[email protected]
www.cs.umb.edu/~eb
University of Florida, Gainesville
March 19, 2002
1
Acknowledgements
• Joint work with (so far)
Victor Guillemin
Tara Holm
• Conversations with
Walter Whiteley
Catalin Zara
and others
• Preprint and slides available at
www.cs.umb.edu/~eb/betti
2
Plan
• Combinatorics  topology  combinatorics
• f vectors, the McMullen conjectures
• Topological ideas for embedded graphs
– Geodesics and connections
– Morse theory and Betti numbers
• McMullen revisited
• Parallel redrawings
• Examples, pretty pictures, open questions
3
Counting faces of a polytope
• Euler: fk = number of faces of dimension k
• Define i by
k
fn-k =

i=0
( n-i
k-i ) i
• McMullen conjectures: For simple polytopes,
i are palindromic and unimodal
• Stanley:
Poincare duality  palindromic
hard Lefshetz theorem  unimodal
4
Dodecahedron
f = (20, 30, 12, 1)
 = (1, 9, 9, 1)
What do the i count?
5
Subject matter
• Connected d-regular graph  embedded in real
Euclidean n-space
• Every pair of edges at a vertex determines a planar
cycle of edges
• These are the geodesics
• 1-skeleton of any simple polytope (since any pair
of edges at a vertex determines a 2-face)
– simplex, cube in any dimension
– dodecahedron, not icosahedron
• More examples from topology …
• More examples not from topology …
6
Johnson graphs J(n,k)
• Vertices are the k element subsets of an n-set
• v,w are adjacent when #(vw) = n-1
{1,2} =
• Represent vertices as bit vectors to
(1,1,0,0)
embed on a hyperplane in n-space
• J(n,1) = Kn (complete graph)
{1,3}
{1,4}
• J(4,2) is the octahedron
• J(n,2) is not the cross polytope
{2,4}
{2,3}
• Topology: Grassmannian
manifold of k-planes in n-space
{3,4}
7
Johnson graph geodesics
• Each pair of edges at a
vertex determines
a geodesic
• Geodesics are
triangles
squares
8
Permutahedra
•
•
•
•
Cayley graphs of the symmetric groups Sn
Vertices are the permutations of an n-set
v,w are adjacent when v w-1 is a transposition
Represent vertices as permutations of (1,…n)
to embed on a hyperplane
in n-space
(1,2,3)
(1,3,2)
• “Internal” edges matter
• S3 is the complete (2,1,3)
bipartite graph
(2,3,1)
K(3,3) in the plane
• Topology: flag manifolds
(3,1,2)
(3,2,1)
9
Geodesics for S3
(1,2,3)
(2,3)
(1,3,2)
(2,1,3)
(1,3)
(1,3)
(2,3)
(1,3)
(2,3,1)
(2,3)
(3,1,2)
(3,2,1)
10
••
• • •
Cayley graph of S4 •
•
•
•
• •
•
••
•
Simplicial geometry and transportation polytopes,
Trans. Amer. Math. Soc. 217 (1976) 138.
11
Geodesics for Sn
(1,2,3,4)
(2,1,3,4)
• Hexagons on S3
slices
• Rectangles
on Klein
4-group
slices
(1,2,4,3)
(2,1,4,3)
12
Betti numbers
i() = number of vertices with down degree i
= ith Betti number

down degree 2
 = (1, m2, 1)
for convex m-gon
1
1
down degree 1
0
When is  = (0, 1, …) independent of  ?
13
… convex not required
 = (2,1,2)
 = (2,2,2)
 = (k, m2k, k) for (convex) m-gon
winding k times (k < m/2, gcd(k,m)=1)
14
… nor need vertices be distinct
 = (2,4,2)
15
… polygon not required
 = (1, 4, 4, 1)
 = (1, 2, 2, 1)
16
… some hypothesis is necessary
 = (1, 2, 1)
 = (2, 0, 2)
17
Inflection free geodesics
• A geodesic is inflection free if it winds
consistently in the same direction in its plane
• All our examples have inflexion free geodesics
18
Betti number invariance
Theorem: Inflection free geodesics
 Betti numbers independent of 
down degrees
v:3, w:2
v
v:2, w:3
w
 Poincare duality (replace  by - )
19
Projections help a lot
• Generic projection to R3 preserves our axioms
• Once you know the geodesics are coplanar in
R3 you can make all Betti number calculations
with a generic plane projection!
20
McMullen reprise
• Theorem: Our Betti numbers are McMullen’s
• Proof: Every k-face has a unique lowest point,
number of down edges at a point determines
the number of k-faces rooted there
2C2 = 1 of these at
each of the 1 = 9
vertices with 2 up
edges
3C2 = 3 of these at
the 0 = 1 vertex
with 3 up edges
21
McMullen reprise
• Betti number invariance implies the first
McMullen conjecture (palindromic)
• With our interpretation of the Betti numbers
how hard can it be to prove they are unimodal?
• Think of our plane pictures as a rotation
invariant Hasse diagram for a poset?
22
Parallel redrawing
• Attach velocity vector to each vertex so that
when the vertices move the new edges are
parallel to the originals
• There are always at least n+1 linearly
independent parallel redrawings:
n translations and the dilation
23
Theorem: A 3-independent embedded graph in Rn
with convex (hence inflection free) geodesics has
n0 + 1 = n + 1
independent parallel redrawings. n+1 of these are
trivial, 11 are interesting.
Proof:
Adapted from Guillemin and Zara
argument in equivariant cohomology of GKM
manifolds
24
Simple convex polytopes
• n 0 +  1 = fn-1 = number of faces
• One parallel redrawing for each face
(includes translations and dilation)
25
More examples
• J(n,2)
 = (1,1,2,2,3,3,4,…,4,3,3,2,2,1,1)
11 = 0, so no nontrivial parallel redrawings
• Symmetric groups
S3
 = (1,2,2,1)
S4
 = (1,3,5,6,5,3,1)
(Mahonian numbers count permutations by number
of inversions)
11 = n 2 nontrivial parallel redrawings
26
Parallel redrawing in the plane
• Parallel redrawings correspond to infinitesimal
motions (rotate velocities 90°)
• Plane m-gon is braced by m3 diagonals, so
has m3+3 = m infinitesimal motions when we
count the rotation and two translations
•  = (k, m2k, k) so we expect 2k+m 2k = m
parallel redrawings when we count the dilation
and the two translations
27
One parallel redrawing for each edge,
whether or not convex or inflection free
dilation and translations are
combinations of these
28
When 3-independence fails
• Need extra awkward hypothesis:
geodesics must be exact
• Suggests parallel redrawing …
29
Desargues’ configuration K2  K3
 = (1, 2, 2, 1), 11 = 1
motion
parallel deformation
(we need the exactness hypothesis)
30
K(3,3)
 = (1, 2, 2, 1)
11 = 1
exactness
 inscribed in conic (converse of Pascal)
 has a motion (Bolker-Roth)
(infinitesimal) motion , parallel deformation
31
The Petersen graph
An exact embedding
with two inflection
free geodesics.
= (1, 4, 4, 1)
6 redrawings
32
Cuboctahedron
Ink on paper. Approximately 8" by 11".
Image copyright (c) 1994 by Andrew Glassner.
33
http://mathworld.wolfram.com/
GreatStellatedDodecahedron.html
Inflection free geodesics
are pentagrams
 = (5, 5, 5, 5)
30 + 1 = 20 = f2,
so behaves as if simple and convex
34
Small Stellated Dodecahedron
http://amath.colorado.edu/appm/staff/fast/
Polyhedra/ssd.html
Inflection free geodesics
are pentagrams and
triangles
= (3,1,2,2,1,3)
Unimodularity fails
35
Great Dodecahedron
36
Great Icosahedron
37
Great Truncated Cuboctahedron
38
Open questions
• Prove the Betti numbers unimodular
• Find the natural boundaries
– Understand the non-3-independent cases
– Understand 0 > 1 (stellations)
• Interpret strange examples topologically
• Make the projective invariance visible
39