{"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/positive-2-bridge-knots-and-chirally-cosmetic","title":"Positive 2-bridge knots and chirally cosmetic surgeries","arxiv_id":"2308.10126","date":"2023-08-19","proceeding":null,"authors":["Michael Huang","Zelong Li","Rahi Tanaz","Chengyi Zhang"],"abstract":"In this paper we verify that with the exception of the $(2, 2n+1)$ torus knots, positive 2-bridge knots up to 31 crossings do not admit chirally cosmetic surgeries. A knot $K$ admits chirally cosmetic surgeries if there exist surgeries $S^3_r$ and $S^3_{r'}$ with distinct slopes $r$ and $r'$ such that $S^3_r(K) \\cong -S^3_{r'}(K)$, where the negative represents an orientation reversal. To verify this, we use the obstruction formula from arXiv:2112.03144 which relates classical knot invariants to the existence of chirally cosmetic surgeries. To check the formula, we develop a Python program that computes the classical knot invariants $a_2$, $a_4$, $v_3$, $\\det$, and $g$ of a positive 2-bridge knot.","url_abs":"https://arxiv.org/abs/2308.10126v2","url_pdf":"https://arxiv.org/pdf/2308.10126v2.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":"positive-2-bridge-knots-and-chirally-cosmetic","repo_url":"https://github.com/zl830/chiral_cosmetic_surgery_for_pos_2_bridge_knots","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}