A Combinatorial Formula for the Bigraded Betti Numbers

Samantha Moore

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arXiv · 2004.02239

A Combinatorial Formula for the Bigraded Betti Numbers

Samantha Moore

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Abstract

It has been shown that $1$-parameter persistence modules have a very simple classification, namely there is a discrete invariant called a barcode that completely characterizes $1$-parameter persistence modules up to isomorphism. In contrast, Carlsson and Zomorodian showed that $n$-parameter persistence modules have no such "nice" classification when $n>1$; every discrete invariant is incomplete. Despite their incompleteness, discrete invariants can still provide insight into the properties of multiparameter persistence modules. A well-studied discrete invariant for $2$-parameter persistence modules is the bigraded Betti numbers. Through commutative algebra techniques, it is known that the bigraded Betti numbers of a $2$-parameter persistence module $M$ can be recovered from the barcodes of certain zigzag modules within $M$ via a simple combinatorial formula. We present an alternate proof of this formula that relies only on basic linear algebra.

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Samantha Moore. 2020-04-05. A Combinatorial Formula for the Bigraded Betti Numbers. https://arxiv.org/abs/2004.02239

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