arXiv subjects
Publications and source records attributed to Oliver Knill.
Analytic torsion is a functional on graphs which only needs linear algebra to be defined. In the continuum it corresponds to the Ray-Singer analytic torsion. We have formulas for analytic torsion if the graph is contractible or if it is a discrete sphere. A key insight is that analytic torsion is the super determinant of the Dirac operator of the graph.
A vexing open problem in information theory is to find the Shannon capacity of odd cyclic graphs larger than the pentagon and especially for the heptagon. Lower bounds for the capacity are obtained by solving King chess puzzles. Upper bounds are obtained by solving entanglement problems, that is to find good Lovasz umbrellas, quantum state realizations of the graph. We observe that optimal states are always pure states. The rest is expository. One general interesting question is whether the Shannon capacity is always some n-th root of the independence number of the n'th power of the graph.
The arithmetic of N, Z, Q, R can be extended to a graph arithmetic where N is the semiring of finite simple graphs and where Z and Q are integral domains, culminating in a Banach algebra R. A single network completes to the Wiener algebra. We illustrate the compatibility with topology and spectral theory. Multiplicative linear functionals like Euler characteristic, the Poincare polynomial or the zeta functions can be extended naturally. These functionals can also help with number theoretical questions. The story of primes is a bit different as the integers are not a unique factorization domain, because there are many additive primes. Most graphs are multiplicative primes.
We show that the curvature K_(G*H)(x,y) at a point (x,y) in the strong product G*H of two arbitrary finite simple graphs is equal to the product K_G(x) K_H(y) of the curvatures.
Discrete d-manifolds are classes of finite simple graphs which can triangulate classical manifolds but which are defined entirely within graph theory. We show that the chromatic number X(G) of a discrete d-manifold G is sandwiched between d+1 and 2(d+1). From the general identity X(A+B)=X(A)+X(B) for the join A+B of two finite simple graphs, it follows that there are (2k)-spheres with chromatic number X=(3k+1) and (2k-1)-spheres with chromatic number X=3k. Examples of 2-manifolds with X(G)=5 have been known since the pioneering work of Fisk. Current data support the that the ceiling function of 3(d+1)/2 could be an upper bound for all d-manifolds G, generalizing a conjecture of Albertson-Stromquist, stating X(G) is bounded above by 5 for all 2-manifolds. For a d-manifold, Fisk has introduced the (d-2)-variety O(G). This graph O(G) has maximal simplices of dimension (d-2) and correspond to complete complete subgraphs K_{d-1} of G for which the dual circle has odd cardinality. In general, O(G) is a union of (d-2)-manifolds. We note that if O(S(x)) is either empty or a (d-3)-sphere for all x then O(G) is a (d-2)-manifold or empty. The knot O(G) is already interesting for 3-manifolds G because Fisk has demonstrated that every possible knot can appear as O(G) for some 3-manifold. For 4-manifolds G especially, the Fisk variety O(G) is a 2-manifold in G as long as all O(S(x)) are either empty or a knot in every unit 3-sphere S(x).
Graph complements G(n) of cyclic graphs are circulant, vertex-transitive, claw-free, strongly regular, Hamiltonian graphs with a Z(n) symmetry, Shannon capacity 2 and known Wiener and Harary index. There is an explicit spectral zeta function and tree or forest data. The forest-tree ratio converges to e. The graphs G(n) are Cayley graphs and so Platonic with isomorphic unit spheres G(n-3)^+, complements of path graphs. G(3d+3) are homotop to wedge sums of two d-spheres and G(3d+2),G(3d+4) are homotop to d-spheres, G(3d+1)^+ are contractible, G(3d+2)^+,G(3d+3)^+ are d-spheres. Since disjoint unions are dual to Zykov joins, graph complements of 1-dimensional discrete manifolds G are homotop to a point, a sphere or a wedge sums of spheres. If the length of every connected component of a 1-manifold is not divisible by 3, the graph complement of G is a sphere. In general, the graph complement of a forest is either contractible or a sphere. All induced strict subgraphs of G(n) are either contractible or homotop to spheres. The f-vectors G(n) or G(n)^+ satisfy a hyper Pascal triangle relation, the total number of simplices are hyper Fibonacci numbers. The simplex generating functions are Jacobsthal polynomials, generating functions of k-king configurations on a circular chess board. While the Euler curvature of circle complements G(n) is constant by symmetry, the discrete Gauss-Bonnet curvature of path complements G(n)^+ can be expressed explicitly from the generating functions. There is now a non-trivial 6 periodic Gauss-Bonnet curvature universality in the complement of Barycentric limits. The Brouwer-Lefschetz fixed point theorem produces a 12-periodicity of the Lefschetz numbers of all graph automorphisms of G(n). There is also a 12-periodicity of Wu characteristic. This is a 4 periodicity in dimension.These are manifestations of stable homotopy features, but combinatorial.
A finite abstract simplicial complex G defines the Barycentric refinement graph phi(G) = (G,{ (a,b), a subset b or b subset a }) and the connection graph psi(G) = (G,{ (a,b), a intersected with b not empty }). We note here that both functors phi and psi from complexes to graphs are invertible on the image (Theorem 1) and that G,phi(G),psi(G) all have the same automorphism group and that the Cartesian product of G corresponding to the Stanley-Reisner product of phi(G) and the strong Shannon product of psi(G), have the product automorphism groups. Second, we see that if G is a Barycentric refinement, then phi(G) and psi(G) are graph homotopic (Theorem 2). Third, if gamma is the geometric realization functor, assigning to a complex or to a graph the geometric realization of its clique complex, then gamma(G) and gamma(phi(G)) and gamma(psi(G)) are all classically homotopic for a Barycentric refined simplicial complex G (Theorem 3). The Barycentric assumption is necessary in Theorem 2 and 3. There is compatibility with Cartesian products of complexes which manifests in the strong graph product of connection graphs: if two graphs A,A' are homotopic and B,B' are homotopic, then A . B is homotopic to A' . B' (Theorem 4) leading to a commutative ring of homotopy classes of graphs. Finally, we note (Theorem 5) that for all simplicial complexes G as well as product G=G_1 x G_2 ... x G_k, the Shannon capacity Theta(psi(G)) of psi(G) is equal to the number m of zero-dimensional sets in G. An explicit Lowasz umbrella in R^m leads to the Lowasz number theta(G) leq m and so Theta(psi(G))=theta(psi(G))=m making Theta compatible with disjoint union addition and strong multiplication.
If h is a ring-valued function on a simplicial complex G we can define two matrices L and g, where the matrix entries are the h energy of homoclinic intersections. We know that the sum over all h values on G is equal to the sum of the Green matrix entries g(x,y). We also have already seen that that the determinants of L or g are both the product of the h(x). In the case where h(x) is the parity of dimension, the sum of the energy values was the standard Euler characteristic and the determinant was a unit. If h(x) was the unit in the ring then L,g are integral quadratic forms which are isospectral and inverse matrices of each other. We prove here that the quadratic energy expression summing over all pairs h(x)^* h(y) of intersecting sets is a signed sum of squares of Green function entries. The quadratic energy expression is Wu characteristic in the case when h is dimension parity. For general h, the quadratic energy expression resembles an Ising Heisenberg type interaction. The conjugate of g is the inverse of L if h takes unit values in a normed ring or in the group of unitary operators in an operator algebra.
We look at connection Laplacians L,g defined by a field h:G to K, where G is a finite set of sets and K is a normed division ring which does not need to be commutative, nor associative but has a conjugation leading to the norm as the square root of h^* h. The target space K can be a normed real division algebra like the quaternions or an algebraic number field like a quadratic field. For parts of the results we can even assume K to be a Banach algebra like an operator algebra on a Hilbert space. The K-valued function h on G then defines connection matrices L,g in which the entries are in K. We show that the Dieudonne determinants of L and g are both equal to the abelianization of the product of all the field values on G. If G is a simplicial complex and h takes values in the units U of K, then g^* is the inverse of L and the sum of the energy values is equal to the sum of the Green function entries g(x,y). If K is the field C of complex numbers, we can study the spectrum of L(G,h) in dependence of the field h. The set of matrices with simple spectrum defines a |G|-dimensional non-compact Kaehler manifold that is disconnected in general and for which we can compute the fundamental group of each connected component.
Positive curvature and bosons Compact positive curvature Riemannian manifolds M with symmetry group G allow Conner-Kobayashi reductions M to N, where N is the fixed point set of the symmetry G. The set N is a union of smaller-dimensional totally geodesic positive curvature manifolds each with even co-dimension. By Berger, N is not empty. By Lefschetz, M and N have the same Euler characteristic. By Frankel, the sum of dimension of any two components in N is smaller than the dimension of M. Reverting the process N to M allows to build up positive curvature manifolds from smaller ones using division algebras and the geodesic flow. From dimension 6 to 24, only four exceptional manifolds have appeared so far, some of them being flag manifolds and related to the special unitary group in three dimensions. We can now draw a periodic system of elements of the known even-dimensional positive curvature manifolds and observe that the list of even-dimensional known positive curvature manifolds has an affinity with the list of known force carriers in physics. Positive mass of the boson matches up with the existence of of two linearly independent harmonic k-forms on the manifold. This motivates to compute more quantities of the exceptional positive curvature manifolds like the lowest non-zero eigenvalues of the Hodge Laplacian L=d d*+d* d or properties of the pairs (u,v) of harmonic 2,4 or 8 forms in the positive mass case.
A theorem of Grove and Searle directly establishes that positive curvature 2d manifolds M with effective circular symmetry group of dimension 8 or less have positive Euler characteristic X(M): the fixed point set N consists of even dimensional positive curvature manifolds and has the Euler characteristic X(N)=X(M). It is not empty by Berger. If N has a co-dimension 2 component, Grove-Searle forces M to be in { RP^2d,S^2d,CP^d }. By Frankel, there can be not two codimension 2 cases. In the remaining cases, Gauss-Bonnet-Chern forces all to have positive Euler characteristic. This simple proof does not quite reach the record 10 or less which uses methods of Wilking but it motivates to analyze the structure of fixed point components N and in particular to look at positive curvature manifolds which admit a U(1) or SU(2) symmetry with connected or almost connected fixed point set N. They have amazing geodesic properties: the fixed point manifold N agrees with the caustic of each of its points and the geodesic flow is integrable. In full generality, the Lefschetz fixed point property X(N)=X(M) and Frankel's dimension theorem dim(M) is less than dim(A) + dim(B) for two different connectivity components A,B of N produce already heavy constraints in building up M from smaller components. It is possible that S^2d, RP^2d, CP^d, HP^d, OP^2, W^6,E^6,W^12,W^24 are actually a complete list of even-dimensional positive curvature manifolds admitting a continuum symmetry. Aside from the projective spaces, the Euler characteristic of the known cases is always 1,2 or 6, where the jump from 2 to 6 happened with the Wallach or Eschenburg manifolds W^6,E^6 which have four fixed point components N=S^2 + S^2 + S^0, the only known case which are not of the Grove-Searle form N=N_1 or N=N_1 + p} with connected N_1.
We look at the functional Y(M) = int_M K(x) dV(x) for compact Riemannian 2d-manifolds M, where K(x) = (2d)! (d!)^-1 (4pi)^-d int_T prod_k=1^d K_t_2k,t_2k+1(x) dt involves products of d sectional curvatures K_ij(x) averaged over the space T sim O(2d) of all orthonormal frames t=(t_1, ... ,t_2d). A discrete version Y_disc(M) with K_d(x) = (d!)^-1 (4pi)^-d sum_sigma prod_k=1^d K_sigma(2k-1),sigma(2k) sums over all permutations sigma of 1,..,2d. Unlike Euler characteristic which by Gauss-Bonnet-Chern is int_M K_GBC dV=X(M), the quantities Y or Y_disc are in general metric dependent. We are interested in Y(M)-X(M) because if M has curvature sign e, then Y(M) e^d and Y_disc(M) are positive while X(M) e^d>0 is only conjectured. We compute Y_disc in a few concrete examples like 2d-spheres, the 4-manifold CP^2, the 6 manifold SO(4) or the 8-manifold SU(3).
Index expectation curvature K(x) = E[i_f(x)] on a compact Riemannian 2d-manifold M is the expectation of Poincare-Hopf indices i_f(x) and so satisfies the Gauss-Bonnet relation that the interval of K over M is Euler characteristic X(M). Unlike the Gauss-Bonnet-Chern integrand, such curvatures are in general non-local. We show that for small 2d-manifolds M with boundary embedded in a parallelizable 2d-manifold N of definite sectional curvature sign e, an index expectation K(x) with definite sign e^d exists. The function K(x) is constructed as a product of sectional index expectation curvature averages K_k(x) = E[i_k(x)] of a probability space of Morse functions f for which i_f(x) is the product of i_k(x), where the i_k are independent and so uncorrelated.
The Hopf sign conjecture states that a compact Riemannian 2d-manifold M of positive curvature has Euler characteristic X(M)>0 and that in the case of negative curvature X(M) (-1)^d >0. The Hopf product conjecture asks whether a positive curvature metric can exist on product manifolds like S^2 x S^2. By formulating curvature integral geometrically, these questions can be explored for finite simple graphs, where it leads to linear programming problems. In this more expository document we aim to explore also a bit of the history of the Hopf conjecture and mention some strategies of attacks which have been tried. We illustrate the new integral theoretic mu-curvature concept by proving that for every positive curvature manifold M there exists a mu-curvature K satisfying Gauss-Bonnet-Chern X(M)=\int_M K dV such that K is positive on an open set U of volume arbitrary close to the volume of M.
We illustrate connections between differential geometry on finite simple graphs G=(V,E) and Riemannian manifolds (M,g). The link is that curvature can be defined integral geometrically as an expectation in a probability space of Poincare-Hopf indices of coloring or Morse functions. Regge calculus with an isometric Nash embedding links then the Gauss-Bonnet-Chern integrand of a Riemannian manifold with the graph curvature. There is also a direct nonstandard approach: if V is a finite set containing all standard points of M and E contains pairs which are infinitesimally close in the sense of internal set theory, one gets a finite simple graph (V,E) which gets a curvature which as a measure corresponds to the standard curvature. The probabilistic approach is an umbrella framework which covers discrete spaces, piecewise linear spaces, manifolds or varieties.
An integral geometric curvature is defined as the index expectation K(x) = E[i(x)] if a probability measure m is given on vector fields on a Riemannian manifold or on a finite simple graph. Such curvatures are local, satisfy Gauss-Bonnet and are independent of any embedding in an ambient space. While realizing constant Gauss-Bonnet-Chern curvature is not possible in general already for 4-manifolds, we prove that for compact connected manifolds, constant curvature K_m can always be realized with m supported on Morse gradient fields. We give examples of finite simple graphs which do not allow for any constant m-curvature and prove that for one-dimensional connected graphs, there is a convex set of constant curvature configurations with dimension of the first Betti number of the graph. In particular, there is always a unique constant curvature solution for trees.
We generalize the Poincare-Hopf theorem sum_v i(v) = X(G) to vector fields on a finite simple graph (V,E) with Whitney complex G. To do so, we define a directed simplicial complex as a finite abstract simplicial complex equipped with a bundle map F: G to V telling which vertex T(x) in x dominates the simplex x. The index i(v) of a vertex v is defined as X(F^-1(v)), where X is the Euler characteristic. We get a flow by adding a section map F: V to G. The resulting map T on G is a discrete model for a differential equation x'=F(x) on a compact manifold. Examples of directed complexes are defined by Whitney complexes defined by digraphs with no cyclic triangles or gradient fields on finite simple graphs defined by a locally injective function. The result extends to simplicial complexes equipped with an energy function H:G to Z that implements a divisor. The index sum is then the total energy.
A finite simple graph G is declared to have positive curvature if every in G embedded wheel graph has five or six vertices. A d-graph is a finite simple graph G for which every unit sphere is a (d-1)-sphere. A d-sphere is a d-graph G for which there exists a vertex x such that G-x is contractible. A graph G is contractible if there is a vertex x such that S(x) and G-x are contractible. The empty graph 0 is the (-1)-sphere. The 1-point graph 1 is contractible. The theorem is that for d bigger than 1, every connected positive curvature d-graph is a d-sphere. A discrete Synge result follows: a positive curvature graph is simply connected and orientable. For every d larger than 1, there are only finitely many positive curvature graphs. There are six for d=2 and all have diameter less or equal to 3. To prove the theorem, we use a "geomag lemma" which shows that every geodesic in G can be extended to an immersed 2-graph S of positive curvature and must so be a 2-sphere with positive curvature. As none of these has diameter larger than 3, also G has a diameter 3 or less. This can be used to show that G-x is contractible and so must be a sphere.