A Combinatorial Formula for the Bigraded Betti Numbers

TL;DR

This paper provides a new, linear algebra-based proof for a combinatorial formula that computes the bigraded Betti numbers of 2-parameter persistence modules, offering insights into their structure despite the lack of a complete classification.

Contribution

It introduces an alternative proof of a known combinatorial formula for bigraded Betti numbers using only basic linear algebra techniques.

Findings

The proof simplifies understanding of bigraded Betti numbers.

It demonstrates the connection between algebraic invariants and zigzag module barcodes.

Provides a more accessible approach to computing invariants in multiparameter persistence.

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.