2020/04/05 by Samantha Moore, Moore, Samantha
Mathematics · #13C05 #55N31 #Algebraic Topology (math.AT) #FOS: Mathematics #math.AT #msc:13C05 #msc:55N31
paper · pdf · doi:10.48550/arxiv.2004.02239
12 pages. Updated to better clarify what was previously known and what aspects are our contributions
arxiv created 2021/10/31 · arxiv updated 2021/11/02
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.