{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-computational-framework-for-weighted","title":"A computational framework for weighted simplicial homology","arxiv_id":"2206.04612","date":"2022-06-09","proceeding":null,"authors":["Andrei C. Bura","Neelav S. Dutta","Thomas J. X. Li","Christian M. Reidys"],"abstract":"We provide a bottom up construction of torsion generators for weighted homology of a weighted complex over a discrete valuation ring $R=\\mathbb{F}[[\\pi]]$. This is achieved by starting from a basis for classical homology of the $n$-th skeleton for the underlying complex with coefficients in the residue field $\\mathbb{F}$ and then lifting it to a basis for the weighted homology with coefficients in the ring $R$. Using the latter, a bijection is established between $n+1$ and $n$ dimensional simplices whose weight ratios provide the exponents of the $\\pi$-monomials that generate each torsion summand in the structure theorem of the weighted homology modules over $R$. We present algorithms that subsume the torsion computation by reducing it to normalization over the residue field of $R$, and describe a Python package we implemented that takes advantage of this reduction and performs the computation efficiently.","url_abs":"https://arxiv.org/abs/2206.04612v1","url_pdf":"https://arxiv.org/pdf/2206.04612v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-computational-framework-for-weighted","repo_url":"https://gitlab.com/nerlf/weightedsimplicialhomology","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}