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How is a graph like a manifold? Ethan Bolker Mathematics UMass-Boston [email protected] www.cs.umb.edu/~eb UMass-Boston September 30, 2002 1 Acknowledgements • Joint work with Victor Guillemin Tara Holm • Conversations with Walter Whiteley Catalin Zara and others • Preprint available 2 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 3 Counting faces of a polytope • Euler: fk = number of faces of dimension k • Define i by fn-k = ( n-i k-i ) i f = (20, 30, 12, 1) = (1, 5, 5, 1) 4 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) 5 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 6 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 7 1-skeleton of a simple polytope Each pair of edges at a vertex determines a 2-face – these are the geodesics 8 The octahedron • Each pair of edges at a vertex lies on a unique geodesic • Geodesics are triangles squares • Each edge belongs to three geodesics • Not simple 9 Johnson graphs J(n,k) • Vertices are the k element subsets of an n-set • v,w are adjacent when #(vw) = k-1 {1,2} = • Represent vertices as bit vectors to (1,1,0,0) embed on a hyperplane in n-space • J(n,1) = Kn {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} 10 Zonohedra • 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) Rhombic dodecahedron: a perspective drawing of the tesseract 11 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 (1,2,3) in n-space (2,1,3) • S3 is the complete (1,3,2) bipartite graph K(3,3) in the plane (2,3,1) • Topology: flag manifolds (3,1,2) (3,2,1) 12 •• • • • Cayley graph of S4 • • • • • • • •• • Simplicial geometry and transportation polytopes, Trans. Amer. Math. Soc. 217 (1976) 138. 13 The Sn connection • Geodesics lie on plane slices corresponding to subgroups • Hexagons come from S3 subgroups (1,*,*,*) (*,1,*,*) • Rectangles come from Klein 4-groups 14 Cuboctahedron Ink on paper. Approximately 8" by 11". Image copyright (c) 1994 by Andrew Glassner. 15 http://mathworld.wolfram.com/ SmallRhombicuboctahedron.html 16 Grea Stellated Dodecahedron 17 Great Icosahedron 18 Great Dodecahedron 19 Great Truncated Cuboctahedron 20 Betti numbers i() = number of vertices with down degree i = ith Betti number down degree 2 = (1, m2, 1) for convex m-gon 1 1 down degree 1 0 When is = (0, 1, 2) independent of ? 21 … convex not required = (2,1,2) = (3,2,3) = (k, m2k, k) for (convex) m-gon winding k times (k < m/2, gcd(k,m)=1) 22 … nor need vertices be distinct = (2,4,2) 23 … polygon not required = (1, 1, 1, 1, 1) S3 = (1, 2, 2, 1) K5 K2 K3 24 … some hypothesis is necessary = (2, 0, 2) = (1, 2, 1) 25 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) 26 Betti number invariance Theorem: Inflection free geodesics Betti numbers independent of down degrees v:3, w:2 v down degrees v:2, w:3 w 27 The Petersen graph 28 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 29 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 ( 22 )= 1 of these at each of the 1 = 9 vertices with 2 up edges ( 32) = 3 of these at • the 0 = 1 vertex with 3 up edges • n-i fn-k = ( k-i ) i 30 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? 31 = (3,1,2,2,1,3) http://amath.colorado.edu/appm/staff/fast/ Polyhedra/ssd.html Small Stellated Dodecahedron 32 = (7, 3, 3, 7) Great Stellated Dodecahedron 33 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) 34 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 35 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 36 Simple polytopes • One parallel redrawing for each face • p = n 0 + 1 = number of faces 37 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 38 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): e v w C(e) 39 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 40 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 41 One parallel redrawing for each edge: dilation and translations are combinations of these 42 Desargues’ configuration = (1, 2, 2, 1), p = 21+2 - 3 = 1 motion parallel deformation (we need the extra hypothesis) 43 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) (infinitesimal) motion , parallel deformation 44 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? 45