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First Betti number of the path homology of random directed graphs
Path homology is a topological invariant for directed graphs, which is sensitive to their asymmetry and can discern between digraphs which are...
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Unboundedness of the first Betti number and the last Betti number of numerical semigroups generated by concatenation
We show that the minimal number of generators and the Cohen-Macaulay type of a family of numerical semigroups generated by concatenation of...
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A bound on the second Betti number of hyperkähler manifolds of complex dimension six
Let M be an irreducible compact hyperkähler manifold of complex dimension six. Under an assumption on the Looijenga–Lunts–Verbitsky decomposition of...
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Nonvanishing Betti numbers of edge ideals of weakly chordal graphs
We show that Kimura’s necessary and sufficient condition for the nonvanishingness of multigraded Betti numbers of edge ideals of chordal graphs can...
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Bounds on the First Betti Number: An Approach via Schatten Norm Estimates on Semigroup Differences
We derive new estimates for the first Betti number of compact Riemannian manifolds. Our approach relies on the Birman–Schwinger principle and...
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Classes of cut ideals and their Betti numbers
We study monomial cut ideals associated to a graph G , which are a monomial analogue of toric cut ideals as introduced by Sturmfels and Sullivant....
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The Betti side of the double shuffle theory. III. Bitorsor structures
In the first two parts of the series, we constructed stabilizer subtorsors of a ‘twisted Magnus’ torsor, studied their relations with the associator...
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Weight spectra of Gabidulin rank-metric codes and Betti numbers
We consider q -matroids and their associated classical matroids derived from Gabidulin rank-metric codes. We express the generalized rank weights of a...
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On the consistency and asymptotic normality of multiparameter persistent Betti numbers
The persistent Betti numbers are used in topological data analysis (TDA) to infer the scales at which topological features appear and disappear in...
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Powers of monomial ideals with characteristic-dependent Betti numbers
We explore the dependence of the Betti numbers of monomial ideals on the characteristic of the field. A first observation is that for a fixed prime p ...
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Graded Betti Numbers of Balanced Simplicial Complexes
We prove upper bounds for the graded Betti numbers of Stanley-Reisner rings of balanced simplicial complexes. Along the way we show bounds for...
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On the Hodge and Betti Numbers of Hyper-Kähler Manifolds
In this survey article, we review past results (obtained by Hirzebruch, Libgober–Wood, Salamon, Gritsenko, and Guan) on Hodge and Betti numbers of...
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Manifolds of positive Ricci curvature, quadratically asymptotically nonnegative curvature, and infinite Betti numbers
In a previous paper (Jiang and Yang (2021)), we constructed complete manifolds of positive Ricci curvature with quadratically asymptotically...