Steinitz Representations L á szl ó Lov á sz Microsoft Research

1. Steinitz Representations László Lovász Microsoft Research One Microsoft Way, Redmond, WA 98052 lovasz@microsoft.com

2. Every 3-connected planar graph is the skeleton of a convex 3-polytope. Steinitz 1922 3-connected planar graph

3. Coin representation Koebe (1936) Every planar graph can be represented by touching circles

4. Polyhedral version Every 3-connected planar graph is the skeleton of a convex polytope such that every edge touches the unit sphere Andre’ev

5. From polyhedra to circles horizon

6. From polyhedra to representation of the dual

7. G: 3-connected planar graph outer face fixed to convex polygon edges replaced by rubber bands Energy: Equilibrium: Rubber bands and planarity Tutte (1963)

8. G 3-connected planar rubber band embedding is planar (Easily) polynomial time computable Lifts to Steinitz representation Maxwell-Cremona Tutte

9. <0, if ijE Mij 0, if weighted adjacency matrix of G G-matrix : eigenvalues of M WLOG G=(V,E): connected graph M=(Mij): symmetric VxV matrix Miiarbitrary

10. G planar, M G-matrix  corank of M is at most 3. Colin de Verdière Van der Holst • G has a K4 or K2,3 minor •  • G-matrix M such that corank of M is 3. Colin de Verdière

11. Proof. (a) True for K4 and K2,3. (b) True for subdivisions of K4 and K2,3. (c) True for graphs containing subdivisions of K4 and K2,3. Induction needs stronger assumption!

12. M transversal intersection symmetric,  X=0 Strong Arnold property VxV symmetric matrices

13. basis of nullspace of M Representation of G in R3 scaling M scaling the ui Nullspace representation

14. connected Van der Holst’s Lemma or… like convex polytopes?

15. Let Mx=0. Then are connected, unless… Van der Holst’s Lemma, restated

16. G 3-connected planar  nullspace representation, scaled to unit vectors, gives embedding in S2 L-Schrijver G 3-connected planar  nullspace representation can be scaled to convex polytope

17. nullspace representation planar embedding

18. x y Equilibrium: Stresses of tensegrity frameworks bars struts cables

19. Bars 0 Cables Braced polyhedra stress-matrix

20. There is no non-zero stress on the edges of a convex polytope Cauchy Every braced polytope has a nowhere zero stress (canonically)

21. q u v p

22. The stress matrix of a nowhere 0 stress on a braced polytope has exactly one negative eigenvalue. The stress matrix of a any stress on a braced polytope has at most one negative eigenvalue. (conjectured by Connelly)

23. Proof:Given a 3-connected planar G, true for • for some Steinitz representation • and the canonical stress; (b) every Steinitz representation and the canonical stress; (c) every Steinitz representation and every stress;

24. Problems • Find direct proof that the canonical • stress matrix has only 1 negative eigenvalue • Directed analog of Steinitz Theorem • recently proved by Klee and Mihalisin. • Connection with eigensubspaces of • non-symmetric matrices?

25. 3. Other eigenvalues? Let . Let span a components; let span b components. Then , unless… From another eigenvalue of the dodecahedron, we get the great star dodecahedron.

26. 4. 4-dimensional analogue? (Colin de Verdière number): maximum corank of a G-matrix with the Strong Arnold property  G planar  G is linklessly embedable in 3-space LL-Schrijver

27. homological, homotopical,… equivalent Apex graph Linklessly embeddable graphs embeddable in R3 without linked cycles

28. G linklessly embeddable  G has no minor in the “Petersen family” Robertson – Seymour - Thomas Basic facts about linklessly embeddable graphs Closed under: - subdivision - minor - Δ-Y and Y- Δ transformations

29. The Petersen family (graphs arising from K6 by Δ-Y and Y- Δ)

30. Given a linklessly embedable graph… Can we construct in P a linkless embedding? Can it be decided in P whether a given embedding is linkless? Is there an embedding that can be certified to be linkless?