The space of barcode bases for persistence modules
The barcode of a persistence module serves as a complete combinatorial invariant of its isomorphism class. Barcodes are typically extracted by performing changes of basis on a persistence module until the constituent matrices have a special form. Here we describe a new algorithm for computing barcod...
| Main Authors: | , , |
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| Format: | Journal article |
| Language: | English |
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Springer
2022
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| _version_ | 1826309882616217600 |
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| author | Jacquard, E Nanda, V Tillmann, U |
| author_facet | Jacquard, E Nanda, V Tillmann, U |
| author_sort | Jacquard, E |
| collection | OXFORD |
| description | The barcode of a persistence module serves as a complete combinatorial invariant of its isomorphism class. Barcodes are typically extracted by performing changes of basis on a persistence module until the constituent matrices have a special form. Here we describe a new algorithm for computing barcodes which also keeps track of, and outputs, such a change of basis. Our main result is an explicit characterisation of the group of transformations that sends one barcode basis to another. Armed with knowledge of the entire space of barcode bases, we are able to show that any map of persistence modules can be represented via a partial matching between bars provided that neither source nor target admits nested bars in its barcode. We also generalise the algorithm and results described above to work for zizag modules. |
| first_indexed | 2024-03-07T07:43:50Z |
| format | Journal article |
| id | oxford-uuid:5ecfa5c2-4821-4a04-be2e-959aba90b0f3 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:43:50Z |
| publishDate | 2022 |
| publisher | Springer |
| record_format | dspace |
| spelling | oxford-uuid:5ecfa5c2-4821-4a04-be2e-959aba90b0f32023-05-16T08:00:34ZThe space of barcode bases for persistence modulesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:5ecfa5c2-4821-4a04-be2e-959aba90b0f3EnglishSymplectic ElementsSpringer2022Jacquard, ENanda, VTillmann, UThe barcode of a persistence module serves as a complete combinatorial invariant of its isomorphism class. Barcodes are typically extracted by performing changes of basis on a persistence module until the constituent matrices have a special form. Here we describe a new algorithm for computing barcodes which also keeps track of, and outputs, such a change of basis. Our main result is an explicit characterisation of the group of transformations that sends one barcode basis to another. Armed with knowledge of the entire space of barcode bases, we are able to show that any map of persistence modules can be represented via a partial matching between bars provided that neither source nor target admits nested bars in its barcode. We also generalise the algorithm and results described above to work for zizag modules. |
| spellingShingle | Jacquard, E Nanda, V Tillmann, U The space of barcode bases for persistence modules |
| title | The space of barcode bases for persistence modules |
| title_full | The space of barcode bases for persistence modules |
| title_fullStr | The space of barcode bases for persistence modules |
| title_full_unstemmed | The space of barcode bases for persistence modules |
| title_short | The space of barcode bases for persistence modules |
| title_sort | space of barcode bases for persistence modules |
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