preprint · arXiv (Cornell University) · 2019

Every 1D Persistence Module is a Restriction of Some Indecomposable 2D\n Persistence Module

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Abstract

A recent work by Lesnick and Wright proposed a visualisation of $2$D\npersistence modules by using their restrictions onto lines, giving a family of\n$1$D persistence modules. We give a constructive proof that any $1$D\npersistence module with finite support can be found as a restriction of some\nindecomposable $2$D persistence module with finite support. As consequences of\nour construction, we are able to exhibit indecomposable $2$D persistence\nmodules whose support has holes as well as an indecomposable $2$D persistence\nmodule containing all $1$D persistence modules with finite support as line\nrestrictions. Finally, we also show that any finite-rectangle-decomposable $n$D\npersistence module can be found as a restriction of some indecomposable\n$(n+1)$D persistence module.\n

Cite this paper

APA 7

Buchet, M., & Escolar, E. G. (2019). Every 1D Persistence Module is a Restriction of Some Indecomposable 2D\n Persistence Module. arXiv (Cornell University). https://doi.org/10.48550/arxiv.1902.07405

MLA 9

Buchet, Mickaël, and Emerson G. Escolar. “Every 1D Persistence Module is a Restriction of Some Indecomposable 2D\n Persistence Module.” arXiv (Cornell University), 2019, https://doi.org/10.48550/arxiv.1902.07405.

Chicago

Buchet, Mickaël, and Emerson G. Escolar. 2019. “Every 1D Persistence Module is a Restriction of Some Indecomposable 2D\n Persistence Module.” arXiv (Cornell University). https://doi.org/10.48550/arxiv.1902.07405.

Harvard

Buchet, M. and Escolar, E.G. (2019) ‘Every 1D Persistence Module is a Restriction of Some Indecomposable 2D\n Persistence Module’, arXiv (Cornell University). Available at: https://doi.org/10.48550/arxiv.1902.07405.

BibTeX

@misc{buchet2019every,
  title = {{Every 1D Persistence Module is a Restriction of Some Indecomposable 2D\n Persistence Module}},
  author = {Buchet, Mickaël and Escolar, Emerson G.},
  journal = {arXiv (Cornell University)},
  year = {2019},
  doi = {10.48550/arxiv.1902.07405}
}

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