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How is a graph like a manifold?

Ethan Bolker

Mathematics

UMass-Boston

eb@cs.umb.edu

www.cs.umb.edu/~eb

UMass-Boston

September 30, 2002

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Acknowledgements• Joint work with

Victor Guillemin Tara Holm

• Conversations with Walter Whiteley Catalin Zara and others

• Preprint available

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Plan

• f vectors, the McMullen conjectures

• Topological ideas for embedded graphs– Geodesics and connections– Lots of examples– Morse theory and Betti numbers

• McMullen revisited

• Examples, open questions, pretty pictures

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Counting faces of a polytope• Euler: fk = number of faces of dimension k

• Define i by

fn-k = ( ) i

n-ik-i

f = (20, 30, 12, 1)

= (1, 5, 5, 1)

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McMullen conjectures (1971)• Simple polytope in Rn:

each vertex has degree n

• For simple polytopes,

i are palindromic and unimodal

• Simplest example is the simplex, a.k.a. Kn+1, the complete graph on n+1 points

= (1, 1, … , 1)

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Stanley’s proof (1980)

• Construct a manifold for which the i

are the Betti numbers

• Poincare duality palindromic

• Hard Lefshetz theorem unimodal

• For Kn+1, the manifold is complex projective n-space

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Connection• A connection on a graph is a set of cycles

(called geodesics) that cover each pair of adjacent edges just once

• Our graphs are always embedded in n-space, and we require that the geodesics be planar

Trivial examples: any plane n-gon

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Each pair of edges at a vertex determines a 2-face – these are the geodesics

1-skeleton of a simple polytope

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The octahedron• Each pair of edges at a

vertex lies on a unique geodesic

• Geodesics aretrianglessquares

• Each edge belongs to three geodesics

• Not simple

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Johnson graphs J(n,k)

• Vertices are the k element subsets of an n-set• v,w are adjacent when #(vw) = k-1• Represent vertices as bit vectors to embed

on a hyperplane in n-space• J(n,1) = Kn

• J(4,2) is the octahedron• J(n,2) is not the cross polytope• Topology: Grassmannian

manifold of k-planes in n-space

{1,2} = (1,1,0,0)

{1,3}

{3,4}

{2,4}

{1,4}

{2,3}

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• Project 1-skeleta of hypercubes (include interior edges)

• Graph is ( ) d

• Geodesics are parallelograms

• In general, products and projections preserve our structures (sections too if done right)

Zonohedra

Rhombic dodecahedron: a perspective drawing of the

tesseract

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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• S3 is the complete bipartite

graph K(3,3) in the plane

• Topology: flag manifolds

(1,2,3)(2,1,3)

(2,3,1)

(3,2,1)(3,1,2)

(1,3,2)

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•

•

• • ••••

••

•

• •

••

Simplicial geometry and transportation polytopes,Trans. Amer. Math. Soc. 217 (1976) 138.

Cayley graph of S4

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The Sn connection(1,*,*,*)

(*,1,*,*)

• Geodesics lie on plane slices corresponding to subgroups

• Hexagons come from S3 subgroups

• Rectangles come from Klein 4-groups

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Cuboctahedron Ink on paper. Approximately 8" by 11".

Image copyright (c) 1994 by Andrew Glassner.

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http://mathworld.wolfram.com/ SmallRhombicuboctahedron.html

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Grea Stellated Dodecahedron

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Great Icosahedron

19Great Dodecahedron

20Great Truncated Cuboctahedron

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Betti numbers i() = number of vertices with down degree i = ith Betti number

1

0

1

down degree 1

down degree 2

= (1, m2, 1) for convex m-gon

When is = (0, 1, 2) independent of ?

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= (k, m2k, k) for (convex) m-gon

winding k times (k < m/2, gcd(k,m)=1)

= (2,1,2)

… convex not required

= (3,2,3)

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… nor need vertices be distinct

= (2,4,2)

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K2 K3

… polygon not required

= (1, 2, 2, 1)

S3 = (1, 1, 1, 1, 1)

K5

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… some hypothesis is necessary

= (1, 2, 1)

= (2, 0, 2)

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Inflection free geodesics

• A geodesic is inflection free if it winds consistently in the same direction in its plane

• All our examples have inflection free geodesics (except the dart)

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Theorem: Inflection free geodesics Betti numbers independent of

down degrees v:3, w:2

Betti number invariance

down degrees v:2, w:3

v w

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The Petersen graph

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Projections help a lot

• Generic projection to R3 preserves our axioms invertibly (projection to the plane makes all geodesics coplanar, so irreversible)

• Once you know the geodesics are coplanar in R3 you can make all Betti number calculations with a generic plane projection

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McMullen reprise• Theorem: Our Betti numbers are McMullen’s

• Proof: Every k-face has a unique lowest point, number of up edges at a point determines the number of k-faces rooted there

fn-k = ( ) in-ik-i

( )= 1 of these at each of the 1 = 9 vertices with 2 up edges

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•

( ) = 3 of these at the 0 = 1 vertex with 3 up edges

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•

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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?

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http://amath.colorado.edu/appm/staff/fast/ Polyhedra/ssd.html Small Stellated Dodecahedron

= (3,1,2,2,1,3)

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Great Stellated Dodecahedron

= (7, 3, 3, 7)

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Open questions

• Find the generalization of convexity that allows you to prove the second McMullen conjecture

• Understand the stellated polytopes• Think of our plane pictures as rotation invariant

Hasse diagrams for a poset?• Understand projective invariance• Explore connections with parallel redrawings

(another talk about things known and unknown)

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Parallel redrawing

• Attach velocity vector to each vertex so that when the vertices move the new edges are parallel to the originals

• There are at least n+1 linearly independent parallel redrawings: the dilation and n translations

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Theorem: (Guillemin and Zara)

An embedded graph in Rn with inflection free geodesics and 0 = 1 has n + 1

independent parallel redrawings.

Proof:

adapted from an argument in equivariant cohomology

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Simple polytopes

• One parallel redrawing for each face

• p = n 0 + 1 = number of faces

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Theorem: Sometimes an embedded graph in Rn has n 0 + 1

independent parallel redrawings. Sometimes it doesn’t.

Challenge: Find the right hypotheses and prove the theorem

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caveat: When more than two edges at a vertex are coplanar, need extra awkward hypothesis: e, C(e) must be a parallel redrawing of (v,w):

v wC(e)

e

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More connections for K4

• Twist standard connection along one edge

• Two geodesics, one of length 3 one of length 9, using some edges twice

• Not inflection free in the plane

• Can twist more edges to make more weird connections

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Examples in the plane

• Parallel redrawings correspond to infinitesimal motions (rotate velocities 90°)

• Plane m-gon is braced by m3 diagonals, so has m3+3 = m infinitesimal motions when we count the rotation and two translations

= (k, m2k, k) so we expect 2k+m 2k = m parallel redrawings when we count the dilation and the two translations

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One parallel redrawing for each edge:

dilation and translations are combinations of these

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motion

parallel deformation

(we need the extra hypothesis)

Desargues’ configuration

= (1, 2, 2, 1), p = 21+2 - 3 = 1

44(infinitesimal) motion , parallel deformation

K(3,3)

= (1, 2, 2, 1)

p = 21+2 - 3 = 1

connection (with extra hypothesis)

inscribed in conic (converse of Pascal)

has a motion (Bolker-Roth)

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Open questions

• Find the natural boundary for the G-Z theorem– Understand the non-3-independent cases– Understand 0 > 1 (stellations)

• Discover meanings for higher Betti numbers• When is a scaffolding a framework?