Publications [#361509] of Woojin Kim

Papers Published

1. Kim, W; Moore, S, Bigraded Betti numbers and Generalized Persistence Diagrams (November, 2021)
   (last updated on 2023/07/05)

   Abstract:
   Commutative diagrams of vector spaces and linear maps over $\mathbb{Z}^2$ are objects of interest in topological data analysis (TDA) where this type of diagrams are called 2-parameter persistence modules. Given that quiver representation theory tells us that such diagrams are of wild type, studying informative invariants of a 2-parameter persistence module $M$ is of central importance in TDA. One of such invariants is the generalized rank invariant, recently introduced by Kim and M\'emoli. Via the M\"obius inversion of the generalized rank invariant of $M$, we obtain a collection of connected subsets $I\subset\mathbb{Z}^2$ with signed multiplicities. This collection generalizes the well known notion of persistence barcode of a persistence module over $\mathbb{R}$ from TDA. In this paper we show that the bigraded Betti numbers of $M$, a classical algebraic invariant of $M$, are obtained by counting the corner points of these subsets $I$s. Along the way, we verify that an invariant of 2-parameter persistence modules called the interval decomposable approximation (introduced by Asashiba et al.) also encodes the bigraded Betti numbers in a similar fashion.