Steinitz Representations László Lovász Microsoft Research One Microsoft Way, Redmond, WA

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

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

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

From polyhedra to circles horizon

From polyhedra to representation of the dual

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

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

G=(V,E) : connected graph M=(M ij ) : symmetric V x V matrix M ii arbitrary M ij < 0, if ij  E 0, if weighted adjacency matrix of G G -matrix : eigenvalues of M WLOG

G planar, M G -matrix  corank of M is at most 3. Colin de Verdière Van der Holst G has a K 4 or K 2,3 minor   G -matrix M such that corank of M is 3. Colin de Verdière

Proof. (a) True for K 4 and K 2,3. (b) True for subdivisions of K 4 and K 2,3. (c) True for graphs containing subdivisions of K 4 and K 2,3. Induction needs stronger assumption!

transversal intersection M V x V symmetric matrices Strong Arnold property symmetric,  X=0

Representation of G in R 3 Nullspace representation basis of nullspace of M scaling M  scaling the u i

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

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

G 3-connected planar  nullspace representation can be scaled to convex polytope G 3-connected planar  nullspace representation, scaled to unit vectors, gives embedding in S 2 L-Schrijver

planar embedding nullspace representation

Stresses of tensegrity frameworks bars struts cables x y Equilibrium:

Cables Braced polyhedra Bars 0 stress-matrix

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

q p uv

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)

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

Problems 1.Find direct proof that the canonical stress matrix has only 1 negative eigenvalue 2.Directed analog of Steinitz Theorem recently proved by Klee and Mihalisin. Connection with eigensubspaces of non-symmetric matrices?

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

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

Linklessly embeddable graphs homological, homotopical,… equivalent embeddable in R 3 without linked cycles Apex graph

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

The Petersen family (graphs arising from K 6 by Δ-Y and Y- Δ)

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