Topology of two-connected graphs and homology of spaces of knots

We propose a new method of computing cohomology groups of spaces of knots in $\R^n$, $n \ge 3$, based on the topology of configuration spaces and two-connected graphs, and calculate all such classes of order $\le 3.$ As a byproduct we define the higher indices, which invariants of knots in $\R^3$ define at arbitrary singular knots. More generally, for any finite-order cohomology class of the space of knots we define its principal symbol, which lies in a cohomology group of a certain finite-dimensional configuration space and characterizes our class modulo the classes of smaller filtration

Counting packings of generic subsets in finite groups

A packing of subsets $\mathcal S_1,..., \mathcal S_n$ in a group $G$ is a sequence $(g_1,...,g_n)$ such that $g_1\mathcal S_1,...,g_n\mathcal S_n$ are disjoint subsets of $G$. We give a formula for the number of packings if the group $G$ is finite and if the subsets $\mathcal S_1,...,\mathcal S_n$ satisfy a genericity condition. This formula can be seen as a generalization of the falling factorials which encode the number of packings in the case where all the sets $\mathcal S_i$ are singletons

Bicircular Matroids with Circuits of at Most Two Sizes

Young in his paper titled, Matroid Designs in 1973, reports that Murty in his paper titled, Equicardinal Matroids and Finite Geometries in 1968, was the first to study matroids with all hyperplanes having the same size. Murty called such a matroid an ``Equicardinal Matroid\u27\u27. Young renamed such a matroid a ``Matroid Design\u27\u27. Further work on determining properties of these matroids was done by Edmonds, Murty, and Young in their papers published in 1972, 1973, and 1970 respectively. These authors were able to connect the problem of determining the matroid designs with specified parameters with results on balanced incomplete block designs. The dual of a matroid design is one in which all circuits have the same size. In 1971, Murty restricted his attention to binary matroids and was able to characterize all connected binary matroids having circuits of a single size. Lemos, Reid, and Wu in 2010, provided partial information on the class of connected binary matroids having circuits of two different sizes. They also shothat there are many such matroids. In general, there are not many results that specify the matroids with circuits of just a few different sizes. Cordovil, Junior, and Lemos provided such results on matroids with small circumference. Here we determine the connected bicircular matroids with all circuits having the same size. We also provide structural information on the connected bicircular matroids with circuits of two different sizes. The bicircular matroids considered are in general non-binary. Hence these results are a start on extending Murty\u27s characterization of binary matroid designs to non-binary matroids