Finite random coverings of one-complexes and the Euler characteristic

This article presents an algebraic topology perspective on the problem of finding a complete coverage probability of a one dimensional domain $X$ by a random covering, and develops techniques applicable to the problem beyond the one dimensional case. In particular we obtain a general formula for the chance that a collection of finitely many compact connected random sets placed on $X$ has a union equal to $X$. The result is derived under certain topological assumptions on the shape of the covering sets (the covering ought to be {\em good}, which holds if the diameter of the covering elements does not exceed a certain size), but no a priori requirements on their distribution. An upper bound for the coverage probability is also obtained as a consequence of the concentration inequality. The techniques rely on a formulation of the coverage criteria in terms of the Euler characteristic of the nerve complex associated to the random covering.Comment: 25 pages,2 figures; final published versio

On volume-preserving vector fields and finite type invariants of knots

We consider the general nonvanishing, divergence-free vector fields defined on a domain in three space and tangent to its boundary. Based on the theory of finite type invariants, we define a family of invariants for such fields, in the style of Arnold's asymptotic linking number. Our approach is based on the configuration space integrals due to Bott and Taubes.Comment: 30 pages, 6 figures, exposition improve

Quantitative Darboux theorems in contact geometry

This paper begins the study of relations between Riemannian geometry and contact topology in any dimension and continues this study in dimension 3. Specifically we provide a lower bound for the radius of a geodesic ball in a contact manifold that can be embedded in the standard contact structure on Euclidean space, that is on the size of a Darboux ball. The bound is established with respect to a Riemannian metric compatible with an associated contact form. In dimension three, it further leads us to an estimate of the size for a standard neighborhood of a closed Reeb orbit. The main tools are classical comparison theorems in Riemannian geometry. In the same context, we also use holomorphic curves techniques to provide a lower bound for the radius of a PS-tight ball.Comment: 33 pages, corrects several inaccuracies in earlier versio

From integrals to combinatorial formulas of finite type invariants -- a case study

We obtain a localized version of the configuration space integral for the Casson knot invariant, where the standard symmetric Gauss form is replaced with a locally supported form. An interesting technical difference between the arguments presented here and the classical arguments is that the vanishing of integrals over hidden and anomalous faces does not require the well-known ``involution tricks''. Further, the integral formula easily yields the well-known arrow diagram expression for the invariant, first presented in the work of Polyak and Viro. We also take the next step of extending the arrow diagram expression to multicrossing knot diagrams and obtain a lower bound for the {\em {\"u}bercrossing number}. The primary motivation is to better understand a connection between the classical configuration space integrals and the arrow diagram expressions for finite type invariants.Comment: 30 (10pt) pages including appendices, 9 figure

On the Reconstruction of Geodesic Subspaces of RN\mathbb{R}^N

We consider the topological and geometric reconstruction of a geodesic subspace of $\mathbb{R}^N$ both from the \v{C}ech and Vietoris-Rips filtrations on a finite, Hausdorff-close, Euclidean sample. Our reconstruction technique leverages the intrinsic length metric induced by the geodesics on the subspace. We consider the distortion and convexity radius as our sampling parameters for a successful reconstruction. For a geodesic subspace with finite distortion and positive convexity radius, we guarantee a correct computation of its homotopy and homology groups from the sample. For geodesic subspaces of $\mathbb{R}^2$, we also devise an algorithm to output a homotopy equivalent geometric complex that has a very small Hausdorff distance to the unknown shape of interest

On the Borsuk conjecture concerning homotopy domination

In the seminal monograph "Theory of retracts", Borsuk raised the following question: suppose two compact ANR's are $h$--equal, i.e. mutually homotopy dominate each other, are they homotopy equivalent? The current paper approaches this question in two ways. On one end, we provide conditions on the fundamental group which guarantee a positive answer to the Borsuk question. On the other end, we construct various examples of compact $h$--equal, not homotopy equivalent continua, with distinct properties. The first class of these examples has trivial all known algebraic invariants (such as homology, homotopy groups etc.) The second class is given by $n$--connected continua, for any $n$, which are infinite $CW$--complexes, and hence ANR's, on a complement of a point.Comment: 18 pages, 6 figures; final version accepted for publicatio