{"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/characterization-and-further-applications-of","title":"Characterization and Further Applications of the Bar-Natan Zh-Construction","arxiv_id":"2307.09387","date":"2023-07-18","proceeding":null,"authors":["Micah Chrisman","Robert G. Todd"],"abstract":"Bar-Natan's Zh-construction associates to each $n$ component virtual link diagram $L$ an $(n+1)$ component virtual link diagram $Zh(L)$. If $L_0,L_1$ are equivalent virtual link diagrams, then $Zh(L_0),Zh(L_1)$ are equivalent as semi-welded links. The importance of the $Zh$-construction is that it unifies several classical knot invariants with virtual knot invariants. For example, the generalized Alexander polynomial of a virtual link diagram $L$ is identical to the usual multi-variable Alexander polynomial of $Zh(L)$. From this it follows that the generalized Alexander polynomial is a slice obstruction: it vanishes on any knot concordant to an almost classical knot. Our main result is a characterization theorem for the $Zh$-construction in terms of almost classical links. Several consequences of this characterization are explored. First, we give a purely geometric description of the $Zh$-construction. Secondly, the $Zh$-construction is used to obtain a simple derivation of the Dye-Kauffman-Miyazawa polynomial. Lastly, we show that every quandle coloring invariant and quandle 2-cocycle coloring invariant can be extended to a new invariant using the $Zh$-construction.","url_abs":"https://arxiv.org/abs/2307.09387v2","url_pdf":"https://arxiv.org/pdf/2307.09387v2.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":"characterization-and-further-applications-of","repo_url":"https://github.com/micah-chrisman/virtual-knots","is_official":1,"mentioned_in_paper":0,"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}