When Left and Right Turns Inside Out: A Geometric and Categorical Introduction to an Inverse Problem in Persistence

TL;DR

This paper explores an inverse problem in persistent homology, focusing on counting embedded spheres in R^3 with specific projection properties and relating geometric isotopies to persistence barcodes.

Contribution

It introduces a novel geometric and categorical framework for an inverse problem in persistence, providing bounds on equivalence classes of embedded spheres based on their persistence barcodes.

Findings

A formula for counting functions with a given barcode

A lower bound on height equivalence classes

A conjectured upper bound on these classes

Abstract

In this paper we introduce the problem of counting embedded spheres in R^3 whose projection to the z-axis yields a level set barcode of a particular type. Two embedded spheres are considered height equivalent if they are related by a z-level set preserving isotopy. A formula previously used to count functions on the interval with the same sub-level set persistence barcode provides a lower bound on height equivalence classes with the same level set persistence. A conjectured upper bound is provided as well.