Local chromatic number of quadrangulations of surfaces

The local chromatic number of a graph G, as introduced in [4], is the minimum integer k such that G admits a proper coloring (with an arbitrary number of colors) in which the neighborhood of each vertex uses less than k colors. In [17] a connection of the local chromatic number to topological properties of (a box complex of) the graph was established and in [18] it was shown that a topological condition implying the usual chromatic number being at least four has the stronger consequence that the local chromatic number is also at least four. As a consequence one obtains a generalization of the following theorem of Youngs [19]: If a quadrangulation of the projective plane is not bipartite it has chromatic number four. The generalization states that in this case the local chromatic number is also four. Both papers [1] and [13] generalize Youngs’ result to arbitrary non-orientable surfaces replacing the condition of the graph being not bipartite by a more technical condition of an odd quadrangulation. This paper investigates when these general results are true for the local chromatic number instead of the chromatic number. Surprisingly, we find out that (unlike in the case of the chromatic number) this depends on the genus of the surface. For the non-orientable surfaces of genus at most four, the local chromatic number of any odd quadrangulation is at least four, but this is not true for non-orientable surfaces of genus 5 or higher. We also prove that face subdivisions of odd quadrangulations and Fisk triangulations of arbitrary surfaces exhibit the same behavior for the local chromatic number as they do for the usual chromatic number

Modified 6j-symbols and 3-manifold invariants

37 pages, 16 figuresInternational audienceWe show that the renormalized quantum invariants of links and graphs in the 3-sphere, derived from tensor categories in ["Modified quantum dimensions and re-normalized link invariants", arXiv:0711.4229] lead to modified 6j-symbols and to new state sum 3-manifold invariants. We give examples of categories such that the associated standard Turaev-Viro 3-manifold invariants vanish but the secondary invariants may be non-zero. The categories in these examples are pivotal categories which are neither ribbon nor semi-simple and have an infinite number of simple objects

Topological invariants from non-restricted quantum groups

We introduce the notion of a relative spherical category. We prove that such a category gives rise to the generalized Kashaev and Turaev-Viro-type 3-manifold invariants defined in arXiv:1008.3103 and arXiv:0910.1624, respectively. In this case we show that these invariants are equal and extend to what we call a relative Homotopy Quantum Field Theory which is a branch of the Topological Quantum Field Theory founded by E. Witten and M. Atiyah. Our main examples of relative spherical categories are the categories of finite dimensional weight modules over non-restricted quantum groups considered by C. De Concini, V. Kac, C. Procesi, N. Reshetikhin and M. Rosso. These categories are not semi-simple and have an infinite number of non-isomorphic irreducible modules all having vanishing quantum dimensions. We also show that these categories have associated ribbon categories which gives rise to re-normalized link invariants. In the case of sl(2) these link invariants are the Alexander-type multivariable invariants defined by Y. Akutsu, T. Deguchi, and T. Ohtsuki.Comment: 37 pages, 16 figure

Hidden Quantum Gravity in 3d Feynman diagrams

In this work we show that 3d Feynman amplitudes of standard QFT in flat and homogeneous space can be naturally expressed as expectation values of a specific topological spin foam model. The main interest of the paper is to set up a framework which gives a background independent perspective on usual field theories and can also be applied in higher dimensions. We also show that this Feynman graph spin foam model, which encodes the geometry of flat space-time, can be purely expressed in terms of algebraic data associated with the Poincare group. This spin foam model turns out to be the spin foam quantization of a BF theory based on the Poincare group, and as such is related to a quantization of 3d gravity in the limit where the Newton constant G_N goes to 0. We investigate the 4d case in a companion paper where the strategy proposed here leads to similar results.Comment: 35 pages, 4 figures, some comments adde

Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations

One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology

Empty Rectangles and Graph Dimension

We consider rectangle graphs whose edges are defined by pairs of points in diagonally opposite corners of empty axis-aligned rectangles. The maximum number of edges of such a graph on $n$ points is shown to be 1/4 n^2 +n -2. This number also has other interpretations: * It is the maximum number of edges of a graph of dimension \bbetween{3}{4}, i.e., of a graph with a realizer of the form \pi_1,\pi_2,\ol{\pi_1},\ol{\pi_2}. * It is the number of 1-faces in a special Scarf complex. The last of these interpretations allows to deduce the maximum number of empty axis-aligned rectangles spanned by 4-element subsets of a set of $n$ points. Moreover, it follows that the extremal point sets for the two problems coincide. We investigate the maximum number of of edges of a graph of dimension $\between{3}{4}$, i.e., of a graph with a realizer of the form \pi_1,\pi_2,\pi_3,\ol{\pi_3}. This maximum is shown to be $1/4 n^2 + O(n)$. Box graphs are defined as the 3-dimensional analog of rectangle graphs. The maximum number of edges of such a graph on $n$ points is shown to be $7/16 n^2 + o(n^2)$