arXiv subjects

Oliver Knill

Publications and source records attributed to Oliver Knill.

At least 91 records · Page 5Linked to original sources

arXiv2017

The strong ring of simplicial complexes

We define a ring R of geometric objects G generated by finite abstract simplicial complexes. To every G belongs Hodge Laplacian H as the square of the Dirac operator determining its cohomology and a unimodular connection matrix L). The sum of the matrix entries of the inverse of L is the Euler characteristic. The spectra of H as well as inductive dimension add under multiplication while the spectra of L multiply. The nullity of the Hodge of H are the Betti numbers which can now be signed. The map assigning to G its Poincare polynomial is a ring homomorphism from R the polynomials. Especially the Euler characteristic is a ring homomorphism. Also Wu characteristic produces a ring homomorphism. The Kuenneth correspondence between cohomology groups is explicit as a basis for the product can be obtained from a basis of the factors. The product in R produces the strong product for the connection graphs and leads to tensor products of connection Laplacians. The strong ring R is also a subring of the full Stanley-Reisner ring S Every element G can be visualized by its Barycentric refinement graph G1 and its connection graph G'. Gauss-Bonnet, Poincare-Hopf or the Brouwer-Lefschetz extend to the strong ring. The isomorphism of R with a subring of the strong Sabidussi ring shows that the multiplicative primes in R are the simplicial complexes and that every connected element in the strong ring has a unique prime factorization. The Sabidussi ring is dual to the Zykov ring, in which the Zykov join is the addition. The connection Laplacian of the d-dimensional lattice remains invertible in the infinite volume limit: there is a mass gap in any dimension.

math.CO↗

arXiv2017

On the arithmetic of graphs

The Zykov ring of signed finite simple graphs with topological join as addition and compatible multiplication is an integral domain but not a unique factorization domain. We know that because by taking graph complements, it becomes isomorphic to the strong Sabidussi ring with disjoint union as addition. We prove that the Euler characteristic is a ring homomorphism from the strong ring to the integers by demonstrating that the strong ring is homotopic to a Stanley-Reisner Cartesian ring. More generally, the Kuenneth formula holds on the strong ring so that the Poincare polynomial is compatible with the ring structure. The Zykov ring has the clique number as a ring homomorphism. Furthermore, the Cartesian ring has the property that the functor which attaches to a graph the spectrum of its connection Laplacian is multiplicative. The reason is that the connection Laplacians do tensor under multiplication, similarly to what the adjacency matrix does for the weak ring. The strong ring product of two graphs contains both the weak and direct product graphs as subgraphs. The Zykov, Sabidussi or Stanley-Reisner rings are so manifestations of a network arithmetic which has remarkable cohomological properties, dimension and spectral compatibility but where arithmetic questions like the complexity of detecting primes or factoring are not yet studied well. We illustrate the Zykov arithmetic with examples, especially from the subring generated by point graphs which contains spheres, stars or complete bipartite graphs. While things are formulated in the language of graph theory, all constructions generalize to the larger category of finite abstract simplicial complexes.

math.CO↗

arXiv2017

On a Dehn-Sommerville functional for simplicial complexes

Assume G is a finite abstract simplicial complex with f-vector (v0,v1, ...), and generating function f(x) = sum(k=1 v(k-1) x^k = v0 x + v1 x^2+ v2 x^3 + ..., the Euler characteristic of G can be written as chi(G)=f(0)-f(-1). We study here the functional f1'(0)-f1'(-1), where f1' is the derivative of the generating function f1 of G1. The Barycentric refinement G1 of G is the Whitney complex of the finite simple graph for which the faces of G are the vertices and where two faces are connected if one is a subset of the other. Let L is the connection Laplacian of G, which is L=1+A, where A is the adjacency matrix of the connection graph G', which has the same vertex set than G1 but where two faces are connected they intersect. We have f1'(0)=tr(L) and for the Green function g L^(-1) also f1'(-1)=tr(g) so that eta1(G) = f1'(0)-f1'(-1) is equal to eta(G)=tr(L-L^(-1). The established formula tr(g)=f1'(-1) for the generating function of G1 complements the determinant expression det(L)=det(g)=zeta(-1) for the Bowen-Lanford zeta function zeta(z)=1/det(1-z A) of the connection graph G' of G. We also establish a Gauss-Bonnet formula eta1(G) = sum(x in V(G1) chi(S(x)), where S(x) is the unit sphere of x the graph generated by all vertices in G1 directly connected to x. Finally, we point out that the functional eta0(G) = sum(x in V(G) chi(S(x)) on graphs takes arbitrary small and arbitrary large values on every homotopy type of graphs.

math.CO↗

arXiv2017

On Helmholtz free energy for finite abstract simplicial complexes

We prove a Gauss-Bonnet formula X(G) = sum_x K(x), where K(x)=(-1)^dim(x) (1-X(S(x))) is a curvature of a vertex x with unit sphere S(x) in the Barycentric refinement G1 of a simplicial complex G. K(x) is dual to (-1)^dim(x) for which Gauss-Bonnet is the definition of Euler characteristic X. Because the connection Laplacian L'=1+A' of G is unimodular, where A' is the adjacency matrix of of the connection graph G', the Green function values g(x,y) = (1+A')^-1_xy are integers and 1-X(S(x))=g(x,x). Gauss-Bonnet for K^+ reads therefore as str(g)=X(G), where str is the super trace. As g is a time-discrete heat kernel, this is a cousin to McKean-Singer str(exp(-Lt)) = X(G) for the Hodge Laplacian L=dd^* +d^*d which lives on the same Hilbert space than L'. Both formulas hold for an arbitrary finite abstract simplicial complex G. Writing V_x(y)= g(x,y) for the Newtonian potential of the connection Laplacian, we prove sum_y V_x(y) = K(x), so that by the new Gauss-Bonnet formula, the Euler characteristic of G agrees with the total potential theoretic energy sum_x,y g(x,y)=X(G) of G. The curvature K now relates to the probability measure p minimizing the internal energy U(p)=sum_x,y g(x,y) p(x) p(y) of the complex. Since both the internal energy (here linked to topology) and Shannon entropy are natural and unique in classes of functionals, we then look at critical points p the Helmholtz free energy F(p)=(1-T) U(p)-T S(p) which combines the energy functional U and the entropy functional S(p)=-sum_x p(x) log(p(x)). As the temperature T changes, we observe bifurcation phenomena. Already for G=K_3 both a saddle node bifurcation and a pitchfork bifurcation occurs. The saddle node bifurcation leads to a catastrophe: the function T -> F(p(T),T) is discontinuous if p(T) is a free energy minimizer.

math.CO↗

arXiv2017

Sphere geometry and invariants

A finite abstract simplicial complex G defines two finite simple graphs: the Barycentric refinement G1, connecting two simplices if one is a subset of the other and the connection graph G', connecting two simplices if they intersect. We prove that the Poincare-Hopf value i(x)=1-X(S(x)), where X is Euler characteristics and S(x) is the unit sphere of a vertex x in G1, agrees with the Green function value g(x,x),the diagonal element of the inverse of (1+A'), where A' is the adjacency matrix of G'. By unimodularity, det(1+A') is the product of parities (-1)^dim(x) of simplices in G, the Fredholm matrix 1+A' is in GL(n,Z), where n is the number of simplices in G. We show that the set of possible unit sphere topologies in G1 is a combinatorial invariant of the complex G. So, also the Green function range of G is a combinatorial invariant. To prove the invariance of the unit sphere topology we use that all unit spheres in G1 decompose as a join of a stable and unstable part. The join operation + renders the category X of simplicial complexes into a monoid, where the empty complex is the 0 element and the cone construction adds 1. The augmented Grothendieck group (X,+,0) contains the graph and sphere monoids (Graphs, +,0) and (Spheres,+,0). The Poincare-Hopf functionals i(G) as well as the volume are multiplicative functions on (X,+). For the sphere group, both i(G) as well as Fredholm characteristic are characters. The join + can be augmented with a product * so that we have a commutative ring (X,+,0,*,1)for which there are both additive and multiplicative primes and which contains as a subring of signed complete complexes isomorphic to the integers (Z,+,0,*,1). We also look at the spectrum of the Laplacian of the join of two graphs. Both for addition + and multiplication *, one can ask whether unique prime factorization holds.

math.GN↗

arXiv2016

On Fredholm determinants in topology

Given an abstract simplicial complex G, the connection graph G' of G has as vertex set the faces of the complex and connects two if they intersect. If A is the adjacency matrix of that connection graph, we prove that the Fredholm characteristic det(1+A) takes values in {-1,1} and is equal to the Fermi characteristic, which is the product of the w(x), where w(x)=(-1)^dim(x). The Fredholm characteristic is a special value of the Bowen-Lanford zeta function and has various combinatorial interpretations. The unimodularity theorem proven here shows that it is a cousin of the Euler characteristic as the later is the sum of the w(x). Unimodularity implies that the matrix 1+A has an inverse which takes integer values. Experiments suggest the conjecture that the range of the Green function values, the union of the entries of the inverse of 1+A form a combinatorial invariant of the simplicial complex and do not change under Barycentric or edge refinements.

math.GN↗

arXiv2016

On Primes, Graphs and Cohomology

The counting function on the natural numbers defines a discrete Morse-Smale complex with a cohomology for which topological quantities like Morse indices, Betti numbers or counting functions for critical points of Morse index are explicitly given in number theoretical terms. The Euler characteristic of the Morse filtration is related to the Mertens function, the Poincaré-Hopf indices at critical points correspond to the values of the Moebius function. The Morse inequalities link number theoretical quantities like the prime counting functions relevant for the distribution of primes with cohomological properties of the graphs. The just given picture is a special case of a discrete Morse cohomology equivalent to simplicial cohomology. The special example considered here is a case where the graph is the Barycentric refinement of a finite simple graph.

math.CO↗

arXiv2016

On Particles and Primes

Primes in the two complete associative normed division algebras C and H have affinities with structures seen in the standard model of particle physics. On the integers in the two algebras, there are two equivalence relations: a strong one, related to a U(1) and SU(3) symmetry allowing to permute and switch signs of the coordinates of the integers, as well as a weak relation with origins from units U(1),SU(2) in the algebra. Weak equivalence classes within the strong equivalence classes of odd primes in C case relate to leptons, the inert ones being neutrino like, and the split ones resembling electron-positron pairs. In the H case, for odd primes, the equivalence classes come in groups of two or three, leading to a caricature of hadrons featuring either mesons built by a quark pair or then baryons obtained by quark triplets. We can now list for every rational prime p all these particles and attach fractional charges to its constituents.

physics.gen-ph↗

arXiv2016

Goldbach for Gaussian, Hurwitz, Octavian and Eisenstein primes

We formulate Goldbach type questions for Gaussian, Hurwitz, Octavian and Eisenstein primes. They are different from Goldbach type statements by Takayoshi Mitsui from 1960 for number fields or C.A. Holben and James Jordan from 1968 for Gaussian integers. Here is what we meeasure: 1) Every even Gaussian integer a+ib satisfying a>2, b>2 is a sum of two Gaussian primes with positive coefficients. 2) Every Eisenstein integer a+bw with a>3,b>3 and w=(1+sqrt(-3))/2 is the sum of two Eisenstein primes with positive coefficients. Note that no evenness condition is asked in the Eisenstein case. 3) Every Lipschitz integer quaternion with positive entries is the sum of two Hurwitz primes. 4) There exists a constant K such that every Octavian integer with coefficients larger than K is the sum of two Octavian primes. Except in the Octonion case, where the fewest experiments were done, the statements can be linked to difficult questions like Landau or Bunyakovsky conjectures. We therefore look also more closely and numerically at some Hardy-Littlewood constants following early computations from Daniel Shanks and Marvin Wunderlich from the 50ies and 70ies.

math.NT↗

arXiv2016

Some experiments in number theory

We experiment with some topics in elementary number theory. For matrices defined by Gaussian primes we observe a circular spectral law for the eigenvalues. We look at matrices defined by Gaussian primes and look at the growth of the determinant, trace. We experiment with various Goldbach conjectures for Gaussian primes, Eisenstein primes, Hurwitz primes or Octavian primes. These conjectures relate with Landau or Bunyakovsky or Andrica type conjectures for rational primes. We also look at some graphs finite simple defined by primes. Finally we look at some spectra of almost periodic pseudo random matrices defined by Diophantine irrational rotations, where fractal spectral phenomena occur. The matrix is the real part of a van der Monde matrix.

math.NT↗

arXiv2016

Gauss-Bonnet for multi-linear valuations

We prove Gauss-Bonnet and Poincare-Hopf formulas for multi-linear valuations on finite simple graphs G=(V,E) and answer affirmatively a conjecture of Gruenbaum from 1970 by constructing higher order Dehn-Sommerville valuations which vanish for all d-graphs without boundary. An example of a quadratic valuation is the Wu characteristic w(G) which sums (-1)^(dim(x)+dim(y)) over all intersecting pairs of complete subgraphs x,y of a G. More generally, an intersection number w(A,B) sums (-1)^(dim(x)+dim(y)) over pairs x,y, where x is in A and y is in B and x,y intersect. w(G) is a quadratic Euler characteristic X(G), where X sums (-1)^dim(x) over all complete subgraphs x of G. We prove that w is multiplicative, like Euler characteristic: w(G x H) = w(G) w(H) for any two graphs and that w is invariant under Barycentric refinements. We construct a curvature K satisfying Gauss-Bonnet w(G) = sum K(a). We also prove w(G) = X(G)-X(dG) for Euler characteristic X which holds for any d-graph G with boundary dG. We also show higher order Poincare-Hopf formulas: there is for every multi-linear valuation X and function f an index i(a) such that sum i(a)=X(G). For d-graphs G and X=w it agrees with the Euler curvature. For the vanishing multi-valuations which were conjectured to exist, like for the quadratic valuation X(G) = (V X) Y with X=(1,-1,1,-1,1),Y=(0,-2,3,-4,5) on 4-graphs, discrete 4 manifolds, where V_{ij}(G) is the f-matrix counting the number of i and j-simplices in G intersecting, the curvature is constant zero. For all graphs and multi-linear Dehn-Sommerville relations, the Dehn-Sommerville curvature K(v) at a vertex is a Dehn-Sommerville valuation on the unit sphere S(v). We show X V(G) Y = v(G) Y for any linear valuation Y of a d-graph G with f-vector v(G). This provides examples for the Gruenbaum conjecture.

cs.DM↗

arXiv2015

Universality for Barycentric subdivision

The spectrum of the Laplacian of successive Barycentric subdivisions of a graph converges exponentially fast to a limit which only depends on the clique number of the initial graph and not on the graph itself. The proof uses an explicit linear operator mapping the clique vector of a graph to the clique vector of the Barycentric refinement. The eigenvectors of its transpose produce integral geometric invariants for which Euler characteristic is one example.

math.SP↗

arXiv2015

A Sard theorem for graph theory

The zero locus of a function f on a graph G is defined as the graph with vertex set consisting of all complete subgraphs of G, on which f changes sign and where x,y are connected if one is contained in the other. For d-graphs, finite simple graphs for which every unit sphere is a d-sphere, the zero locus of (f-c) is a (d-1)-graph for all c different from the range of f. If this Sard lemma is inductively applied to an ordered list functions f_1,...,f_k in which the functions are extended on the level surfaces, the set of critical values (c_1,...,c_k) for which F-c=0 is not a (d-k)-graph is a finite set. This discrete Sard result allows to construct explicit graphs triangulating a given algebraic set. We also look at a second setup: for a function F from the vertex set to R^k, we give conditions for which the simultaneous discrete algebraic set { F=c } defined as the set of simplices of dimension in {k, k+1,...,n} on which all f_i change sign, is a (d-k)-graph in the barycentric refinement of G. This maximal rank condition is adapted from the continuum and the graph { F=c } is a (n-k)-graph. While now, the critical values can have positive measure, we are closer to calculus: for k=2 for example, extrema of functions f under a constraint {g=c} happen at points, where the gradients of f and g are parallel D f = L D g, the Lagrange equations on the discrete network. As for an application, we illustrate eigenfunctions of geometric graphs and especially the second eigenvector of 3-spheres, which by Courant-Fiedler has exactly two nodal regions. The separating nodal surface of the second eigenfunction f_2 belonging to the smallest nonzero eigenvalue always appears to be a 2-sphere in experiments if G is a 3-sphere.

cs.DM↗

arXiv2015

The graph spectrum of barycentric refinements

Given a finite simple graph G, let G' be its barycentric refinement: it is the graph in which the vertices are the complete subgraphs of G and in which two such subgraphs are connected, if one is contained into the other. If L(0)=0 = dim(G). Let G(m) be the sequence of barycentric refinements of G=G(0). We prove that for any finite simple graph G, the spectral functions F(G(m)) of successive refinements converge for m to infinity uniformly on compact subsets of (0,1) and exponentially fast to a universal limiting eigenvalue distribution function F which only depends on the clique number respectively the dimension d of the largest complete subgraph of G and not on the starting graph G. In the case d=1, where we deal with graphs without triangles, the limiting distribution is the smooth function F(x) = 4 sin^2(pi x/2). This is related to the Julia set of the quadratic map T(z) = 4z-z^2 which has the one dimensional Julia set [0,4] and F satisfies T(F(k/n))=F(2k/n) as the Laplacians satisfy such a renormalization recursion. The spectral density in the d=1 case is then the arc-sin distribution which is the equilibrium measure on the Julia set. In higher dimensions, where the limiting function F still remains unidentified, F' appears to have a discrete or singular component.

cs.DM↗

arXiv2015

The Jordan-Brouwer theorem for graphs

We prove a discrete Jordan-Brouwer-Schoenflies separation theorem telling that a (d-1)-sphere H embedded in a d-sphere G defines two different connected graphs A,B in G such a way that the intersection of A and B is H and the union is G and such that the complementary graphs A,B are both d-balls. The graph theoretic definitions are due to Evako: the unit sphere of a vertex x of a graph G=(V,E) is the graph generated by {y | (x,y) in E} Inductively, a finite simple graph is called contractible if there is a vertex x such that both its unit sphere S(x) as well as the graph generated by V-{x} are contractible. Inductively, still following Evako, a d-sphere is a finite simple graph for which every unit sphere is a (d-1)-sphere and such that removing a single vertex renders the graph contractible. A d-ball B is a contractible graph for which each unit sphere S(x) is either a (d-1)-sphere in which case x is called an interior point, or S(x) is a (d-1)-ball in which case x is called a boundary point and such that the set of boundary point vertices generates a (d-1)-sphere. These inductive definitions are based on the assumption that the empty graph is the unique (-1)-sphere and that the one-point graph K_1 is the unique 0-ball and that K_1 is contractible. The theorem needs the following notion of embedding: a sphere H is embedded in a graph G if it is a sub-graph of G and if any intersection with any finite set of mutually adjacent unit spheres is a sphere. A knot of co-dimension k in G is a (d-k)-sphere H embedded in a d-sphere G.

cs.DM↗

arXiv2015

The Kuenneth formula for graphs

We construct a Cartesian product G x H for finite simple graphs. It satisfies the Kuenneth formula: H^k(G x H) is a direct sum of tensor products H^i(G) x H^j(G) with i+j=k and so p(G x H,x) = p(G,x) p(H,y) for the Poincare polynomial p(G,x) and X(G x H) = X(G) X(H) for the Euler characteristic X(G)=p(G,-1). G1=G x K1 has as vertices the simplices of G and a natural digraph structure. We show that dim(G1) is larger or equal than dim(G) and G1 is homotopic to G. The Kuenneth identity is proven using Hodge describing the harmonic forms by the product f g of harmonic forms of G and H and uses a discrete de Rham theorem given by a combinatorial chain homotopy between simplicial and de Rham cohomology. We show dim(G x H) = dim(G1) + dim(H1) implying that dim(G x H) is larger or equal than dim(G) + dim(H) as for Hausdorff dimension in the continuum. The chromatic number c(G1) is smaller or equal than c(G) and c(G x H) is bounded above by c(G)+c(H)-1. The automorphism group of G x H contains Aut(G) x Aut(H). If G~H and U~V then (G x U) ~ (H x V) if ~ means homotopic: homotopy classes can be multiplided. If G is k-dimensional geometric meaning that all unit spheres S(x) in G are (k-1)-discrete homotopy spheres, then G1 is k-dimensional geometric. If G is k-dimensional geometric and H is l-dimensional geometric, then G x H is geometric of dimension (l+k). The product extends to a ring of chains which unlike the category of graphs is closed under boundary operation taking quotients G/A with A subset Aut(G). As we can glue graphs or chains, joins or fibre bundles can be defined with the same features as in the continuum, allowing to build isomorphism classes of bundles.

math.CO↗

arXiv2015

Graphs with Eulerian unit spheres

d-spheres in graph theory are inductively defined as graphs for which all unit spheres S(x) are (d-1)-spheres and that the removal of one vertex renders the graph contractible. Eulerian d-spheres are geometric d-spheres which are d+1 colorable. We prove here that G is an Eulerian sphere if and only if the degrees of all the (d-2)-dimensional sub-simplices in G are even. This generalizes a Kempe-Heawood result for d=2 and is work related to the conjecture that all d-spheres have chromatic number d+1 or d+2 which is based on the geometric conjecture that every d-sphere can be embedded in an Eulerian (d+1)-sphere. For d=2, such an embedding into an Eulerian 3-sphere would lead to a geometric proof of the 4 color theorem, allowing to see "why 4 colors suffice". To achieve the goal of coloring a d-sphere G with d+2 colors, we hope to embed it into a (d+1)-sphere and refine or thin out the later using special homotopy deformations without touching the embedded sphere. Once rendered Eulerian and so (d+2)-colorable, it colors the embedded graph G. In order to define the degree of a simplex, we introduce a notion of dual graph H' of a subgraph H in a general finite simple graph G. This leads to a natural sphere bundle over the simplex graph. We look at geometric graphs which admit a unique geodesic flow: their unit spheres must be Eulerian. We define Platonic spheres graph theoretically as d-spheres for which all unit spheres S(x) are graph isomorphic Platonic (d-1)-spheres. Gauss-Bonnet allows a classification within graph theory: all spheres are Platonic for d=1, the octahedron and icosahedron are the Platonic 2-spheres, the sixteen and six-hundred cells are the Platonic 3-spheres. The cross polytop is the unique Platonic d-sphere for d>3. It is Eulerian.

math.CO↗

arXiv2014

Coloring graphs using topology

Higher dimensional graphs can be used to colour two-dimensional geometric graphs. If G the boundary of a three dimensional graph H for example, we can refine the interior until it is colourable with 4 colours. The later goal is achieved if all interior edge degrees are even. Using a refinement process which cuts the interior along surfaces we can adapt the degrees along the boundary of that surface. More efficient is a self-cobordism of G with itself with a host graph discretizing the product of G with an interval. It follows from the fact that Euler curvature is zero everywhere for three dimensional geometric graphs, that the odd degree edge set O is a cycle and so a boundary if H is simply connected. A reduction to minimal colouring would imply the four colour theorem. The method is expected to give a reason "why 4 colours suffice" and suggests that every two dimensional geometric graph of arbitrary degree and orientation can be coloured by 5 colours: since the projective plane can not be a boundary of a 3-dimensional graph and because for higher genus surfaces, the interior H is not simply connected, we need in general to embed a surface into a 4-dimensional simply connected graph in order to colour it. This explains the appearance of the chromatic number 5 for higher degree or non-orientable situations, a number we believe to be the upper limit. For every surface type, we construct examples with chromatic number 3,4 or 5, where the construction of surfaces with chromatic number 5 is based on a method of Fisk. We have implemented and illustrated all the topological aspects described in this paper on a computer. So far we still need human guidance or simulated annealing to do the refinements in the higher dimensional host graph.

math.CO↗