Heegaard-Floer homology and string links

We extend knot Floer homology to string links in D^{2} \times I and to d-based links in arbitrary three manifolds, without any hypothesis on the null-homology of the components. As for knot Floer homology we obtain a description of the Euler characteristic of the resulting homology groups (in D^{2} \times I) in terms of the torsion of the string link. Additionally, a state summation approach is described using the equivalent of Kauffman states. Furthermore, we examine the situtation for braids, prove that for alternating string links the Euler characteristic determines the homology, and develop similar composition formulas and long exact sequences as in knot Floer homology.Comment: 57 page

Six signed Petersen graphs, and their automorphisms

Up to switching isomorphism there are six ways to put signs on the edges of the Petersen graph. We prove this by computing switching invariants, especially frustration indices and frustration numbers, switching automorphism groups, chromatic numbers, and numbers of proper 1-colorations, thereby illustrating some of the ideas and methods of signed graph theory. We also calculate automorphism groups and clusterability indices, which are not invariant under switching. In the process we develop new properties of signed graphs, especially of their switching automorphism groups.Comment: 39 pp., 7 fi

3-manifolds efficiently bound 4-manifolds

It is known since 1954 that every 3-manifold bounds a 4-manifold. Thus, for instance, every 3-manifold has a surgery diagram. There are several proofs of this fact, including constructive proofs, but there has been little attention to the complexity of the 4-manifold produced. Given a 3-manifold M of complexity n, we show how to construct a 4-manifold bounded by M of complexity O(n^2). Here we measure ``complexity'' of a piecewise-linear manifold by the minimum number of n-simplices in a triangulation. It is an open question whether this quadratic bound can be replaced by a linear bound. The proof goes through the notion of "shadow complexity" of a 3-manifold M. A shadow of M is a well-behaved 2-dimensional spine of a 4-manifold bounded by M. We prove that, for a manifold M satisfying the Geometrization Conjecture with Gromov norm G and shadow complexity S, c_1 G <= S <= c_2 G^2 for suitable constants c_1, c_2. In particular, the manifolds with shadow complexity 0 are the graph manifolds.Comment: 39 pages, 21 figures; added proof for spin case as wel

The submonoid and rational subset membership problems for graph groups

We show that the membership problem in a finitely generated submonoid of a graph group (also called a right-angled Artin group or a free partially commutative group) is decidable if and only if the independence graph (commutation graph) is a transitive forest. As a consequence we obtain the first example of a finitely presented group with a decidable generalized word problem that does not have a decidable membership problem for finitely generated submonoids. We also show that the rational subset membership problem is decidable for a graph group if and only if the independence graph is a transitive forest, answering a question of Kambites, Silva, and the second author. Finally we prove that for certain amalgamated free products and HNN-extensions the rational subset and submonoid membership problems are recursively equivalent. In particular, this applies to finitely generated groups with two or more ends that are either torsion-free or residually finite

Topics in social network analysis and network science

This chapter introduces statistical methods used in the analysis of social networks and in the rapidly evolving parallel-field of network science. Although several instances of social network analysis in health services research have appeared recently, the majority involve only the most basic methods and thus scratch the surface of what might be accomplished. Cutting-edge methods using relevant examples and illustrations in health services research are provided

A topological comparison of surface extraction algorithms

In many application areas, it is useful to convert the discrete information stored in the nodes of a regular grid into a continuous boundary model. Isosurface extraction algorithms di er on how the discrete information in the grid is generated, on what information does the grid store and on the properties of the output surface.Preprin

A topological comparison of surface extraction algorithms

In many application areas, it is useful to convert the discrete information stored in the nodes of a regular grid into a continuous boundary model. Isosurface extraction algorithms differ on how the discrete information in the grid is generated, on what information does the grid store and on the properties of the output surface. Recent algorithms offer different solutions for the disambiguation problem and for controlling the final topology. Based on a number of properties of the grid’s grey cells and of the reconstruction algorithms, a characterization of several surface extraction strategies is proposed. The classification presented shows the inherent limitations of the different algorithms concerning global topology control and reconstruction of local features like thin portions of the volume and almost non-manifold regions. These limitations can be observed and are illustrated with some practical examples. We review in light of this classification some of the relevant papers in the literature, and see that they cluster in some areas of the proposed hierarchy, making a case for where it might be more interesting to focus in future research.Preprin