{"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/perturbed-traceless-su-2-character-varieties","title":"Perturbed Traceless SU(2) Character Varieties of Tangle Sums","arxiv_id":"2412.06066","date":"2024-12-08","proceeding":null,"authors":["Kai Smith"],"abstract":"If a link $L$ can be decomposed into the union of two tangles $T\\cup_{S^2} S$ along a 2-sphere intersecting $L$ in 4 points, then the intersections of perturbed traceless SU(2) character varieties of tangles in a space called the pillowcase form a set of generators for Kronheimer and Mrowka's reduced singular instanton homology, $I^\\natural$. It is conjectured by Cazassus, Herald, Kirk, and Kotelskiy that with the addition of bounding cochains, the differential of $I^\\natural$ can be recovered from these Lagrangians as well. This article gives a method to compute the perturbed character variety for a large class of tangles using cut-and-paste methods. In particular, given two tangles, $T$ and $S$, Conway defines the tangle sum $T+S$. Given the character varieties of $T$ and $S$, we show how to construct the perturbed character variety of $T+S$. This is done by first studying the perturbed character variety of a certain tangle $C_3$ properly embedded in $S^3$ with 3 balls removed. Using these results, we prove a nontriviality result for the bounding cochains in the conjecture of Cazassus, Herald, Kirk, and Kotelskiy.","url_abs":"https://arxiv.org/abs/2412.06066v1","url_pdf":"https://arxiv.org/pdf/2412.06066v1.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":"perturbed-traceless-su-2-character-varieties","repo_url":"https://github.com/kaismith/pcase","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}