We study an important preprocessing step for the efficient calculation of persistent homology: coarsening a set of points while controlling the quality of the induced persistence diagram. This coarsening step is instrumental in reducing the overall runtime of state-of-the-art algorithms such as Ripser...
We prove a connected sum formula for involutive Heegaard Floer homology, and use it to study the involutive correction terms of connected sums. In particular, we give an example of a three-manifold with \\(\\underline{d}(Y) e d(Y) e \\bar{d}(Y)\\) . We also construct a ...
Connected components in the intersection of two open opposite Schubert cells in real complete flag manifold The crucial step of our reduction uses the parametrization of the space of real unipotent totally positive upper triangular matrices introduced by Lusztig and... B Shapiro,M Shapiro,A Vain...
Using this technique we prove that the Morse-Novikov number of the knot nC (the connected sum of n copies of the Conway knot) is not less than 2n/5 for every positive integer n. We prove also that MN(nC) is not greater than 2n. The same estimates hold for the Morse-Novikov numbers...
The present paper faces the problem of computing the minimal second Betti number of compact 4-manifolds bounded by a given closed 3-manifold M. Note that if the bounding 4-manifolds are assumed to be simply connected, then the problem may be solved via the Boyer classification theorem [S. ...
These spaces are connected by boundary operators \(\partial _k:C_k(K) \rightarrow C_{k-1}(K)\) mapping each k-simplex \(\sigma\) in the sum of the \((k-1)\)-simplices of K strictly contained in \(\sigma\). We denote as \(Z_k(K):=\ker \partial _k\) the space of...
The rank of the zero-dimensional homology group \(H_{0} \left( X \right)\) counts the number of connected components, the rank of the one-dimensional homology group \(H_{1} \left( X \right)\) counts the number of holes and so on. These ranks of homology groups are also known ...
The magnitude of a graph, defined as $\#G=\sum(Z_{G}(q))^{-1}(x,y)\in Q(q)$ for $x,y\in G$, is an integer power series and is an algebraic invariant associated to a graph. Hepworth and Willerton introduced a bi-graded homology theory for graph which has magnitude as its...
In the case of quadratic singularities, we use the results of the calculations to give a topological description (as specific as possible) of such a hypersurface by means of decomposing it into a connected sum. In this case the topological type of the hypersurface is determined by its ...
Lis compact, connected and relatively exact, i.e.vanishes on the relative homotopy group. 3. is compactly supported for alltand constant intnear 0 and 1. The first two assumptions constrain the behaviour of the holomorphic curves that we will consider. Note that givennot satisfying the third,...