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Identifying cobordisms using kernel persistence

Yossi Bokor Bleile, Lisbeth Fajstrup, Teresa Heiss, Anne Marie Svane, Søren Strandskov Sørensen

TL;DR

The paper addresses identifying tunnels, modeled as open cobordisms, between disjoint subcomplexes in a filtered cell complex, with motivation from material science to detect atomic tunnels in point clouds. It introduces a homological construction via a map $Φ$ from $Ker(iota^A_*) ⊕ Ker(iota^B_*)$ to $Ker(iota^{A∪B}_*)$ and treats $Cok(Φ)$ as a persistence module to capture birth and death events of cobordisms along the filtration. Two main algorithmic contributions are presented: (i) kernel persistence to compute birth times in the relevant kernel modules, and (ii) a pairing procedure that combines these to identify birth–death pairs in $Cok(Φ)$ and to produce explicit cobordism representatives. The framework enables systematic detection and characterization of tunnel-like voids in atomic structures from point clouds, with extensions proposed to higher-order cobordisms, a Voronoi-dual formulation, and handling of practical issues such as slice thickness and periodic boundaries.

Abstract

Motivated by applications in chemistry, we give a homlogical definition of tunnels, or more generally cobordisms, connecting disjoint parts of a cell complex. For a filtered complex, this defines a persistence module. We give a method for identifying birth and death times using kernel persistence and a matrix reduction algorithm for pairing birth and death times.

Identifying cobordisms using kernel persistence

TL;DR

The paper addresses identifying tunnels, modeled as open cobordisms, between disjoint subcomplexes in a filtered cell complex, with motivation from material science to detect atomic tunnels in point clouds. It introduces a homological construction via a map from to and treats as a persistence module to capture birth and death events of cobordisms along the filtration. Two main algorithmic contributions are presented: (i) kernel persistence to compute birth times in the relevant kernel modules, and (ii) a pairing procedure that combines these to identify birth–death pairs in and to produce explicit cobordism representatives. The framework enables systematic detection and characterization of tunnel-like voids in atomic structures from point clouds, with extensions proposed to higher-order cobordisms, a Voronoi-dual formulation, and handling of practical issues such as slice thickness and periodic boundaries.

Abstract

Motivated by applications in chemistry, we give a homlogical definition of tunnels, or more generally cobordisms, connecting disjoint parts of a cell complex. For a filtered complex, this defines a persistence module. We give a method for identifying birth and death times using kernel persistence and a matrix reduction algorithm for pairing birth and death times.

Paper Structure

This paper contains 12 sections, 4 theorems, 8 equations, 6 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. A homological way of representing cobordisms
  3. Definition of $\Phi$ in terms of relative homology
  4. More on the point cloud case
  5. $\operatorname{Cok}\left(\Phi\right)$ as persistence module
  6. Births and deaths in $\operatorname{Cok}\left(\Phi\right)$
  7. Algorithms
  8. The algorithm for finding kernel persistence
  9. Algorithm for pairing births and deaths in $\operatorname{Cok}\left(\Phi\right)$
  10. Discussion of representatives
  11. Discussion and future directions
  12. The Voronoi dual filtered complex

Key Result

Theorem 3.1

A class in $\operatorname{Cok}\left(\Phi\right)$ is born when a class in $\operatorname{Ker}\left(\iota^{\mathbb{A}\cup\mathbb{B}}_{\ast}\right)$ is born and there is no class born in $\operatorname{Ker}\left(\iota^{\mathbb{A}}_{\ast}\right)$ and $\operatorname{Ker}\left(\iota^{\mathbb{B}}_{\ast}\ri

Figures (6)

  • Figure 1: Left: tunnel through a point cloud connecting top and bottom. Right: Balls placed around points. The tunnel now corresponds to a path in the complement connecting top and bottom
  • Figure 2: The triangulated cylinder from Example \ref{['ex:cylinder']}.
  • Figure 3: Examples of tunnels. The blue surface is $\mathbb{X}$, the part in the top slice is $\mathbb{A}$ and the part in the bottom slice is $\mathbb{B}$. In a), the tunnel is closed in the middle and hence not counted because it is a sum of two elements in $\operatorname{Ker}\left(\iota_*^\mathbb{A}\right)$ and $\operatorname{Ker}\left(\iota_*^\mathbb{B}\right)$. In b), there is 1 tunnel, and in c) and d), there are no tunnels since these are elements of $\operatorname{Ker}\left(\iota_*^\mathbb{A}\right)$.
  • Figure 4: Examples of cobordisms in $\operatorname{Cok}\left(\Phi_1\right)$. The blue surface is $\mathbb{X}$, the part in the top slice is $\mathbb{A}$ and the part in the bottom slice is $\mathbb{B}$. The typical case is a) where the tunnel exits through the top and bottom. In both b) and c) we count 1 tunnel, while d) is not counted as a tunnel.
  • Figure 5: The 8 cases in Table \ref{['tab:birth-death-events']}. The cell that is inserted when a birth or death happens is shown in green.
  • ...and 1 more figures

Theorems & Definitions (11)

  • Definition 1: Cobordisms from $\mathbb{A}$ to $\mathbb{B}$
  • Example 1
  • Example 2
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • Corollary 4.2
  • proof
  • Theorem A.1
  • ...and 1 more