In this article, we give explicit minimal generators of the first syzygy Hibi ring for a planar distributive lattice in terms sublattices. We also characterization when it is linearly related and derive an exact formula Betti number lattice.

The Betti numbers are fundamental topological quantities that describe the k-dimensional connectivity of an object: beta{0} is the number of connected components and beta{k} effectively counts the number of k-dimensional holes. Although they are appealing natural descriptors of shape, the higher-order Betti numbers are more difficult to compute than other measures and so have not previously bee...

Betti numbers of configuration spaces of mechanical linkages (known also as polygon spaces) depend on a large number of parameters – the lengths of the bars of the linkage. Motivated by applications in topological robotics, statistical shape theory and molecular biology, we view these lengths as random variables and study asymptotic values of the average Betti numbers as the number of links n t...

We study the stable norm on the first homology of a closed, non-orientable surface equipped with a Riemannian metric. We prove that in every conformal class there exists a metric whose stable norm is polyhedral. Furthermore the stable norm is never strictly convex if the first Betti number of the surface is greater than two.

We already showed that Betti numbers are all zero when w is not equal to h for (w,h)-doubly weighted homology groups of some special pre Lie superalgebra and the first number 0 = h. In this note, we show second

We consider the open problem of determining the graded Betti numbers for fat point subschemes Z supported at general points of P. We relate this problem to the open geometric problem of determining the splitting type of the pullback of ΩP2 to the normalization of certain rational plane curves. We give a conjecture for the graded Betti numbers which would determine them in all degrees but one fo...

Let R be a real closed field. The problem of obtaining tight bounds on the Betti numbers of semi-algebraic subsets of R in terms of the number and degrees of the defining polynomials has been an important problem in real algebraic geometry with the first results due to Olĕınik and Petrovskĭı, Thom and Milnor. These bounds are all exponential in the number of variables k. Motivated by several ap...

Let $ Q be a closed manifold with non-trivial first Betti number that admits S^1 $-action, and \Sigma \subseteq T^*Q non-degenerate starshaped hypersurface. We prove the of geometrically distinct Reeb orbits period at most T on grows least logarithmically in $.