{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:38:26Z","timestamp":1760236706732,"version":"build-2065373602"},"reference-count":25,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,12,23]],"date-time":"2021-12-23T00:00:00Z","timestamp":1640217600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>F-polynomials for virtual knots were defined by Kaur, Prabhakar and Vesnin in 2018 using flat virtual knot invariants. These polynomials naturally generalize Kauffman\u2019s affine index polynomial and use smoothing in the classical crossing of a virtual knot diagram. In this paper, we introduce weight functions for ordered orientable virtual and flat virtual links. A flat virtual link is an equivalence class of virtual links with respect to a local symmetry changing a type of classical crossing in a diagram. By considering three types of smoothing in classical crossings of a virtual link diagram and suitable weight functions, there is provided a recurrent construction for new invariants. It is demonstrated by explicit examples that newly defined polynomial invariants are stronger than F-polynomials.<\/jats:p>","DOI":"10.3390\/sym14010015","type":"journal-article","created":{"date-parts":[[2021,12,23]],"date-time":"2021-12-23T21:40:21Z","timestamp":1640295621000},"page":"15","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Recurrent Generalization of F-Polynomials for Virtual Knots and Links"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0264-506X","authenticated-orcid":false,"given":"Amrendra","family":"Gill","sequence":"first","affiliation":[{"name":"Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar 140001, Punjab, India"}]},{"given":"Maxim","family":"Ivanov","sequence":"additional","affiliation":[{"name":"Laboratory of Topology and Dynamics, Novosibirsk State University, 630090 Novosibirsk, Russia"}]},{"given":"Madeti","family":"Prabhakar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar 140001, Punjab, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7553-1269","authenticated-orcid":false,"given":"Andrei","family":"Vesnin","sequence":"additional","affiliation":[{"name":"Regional Scientific and Educational Mathematical Center, Tomsk State University, 634050 Tomsk, Russia"},{"name":"Faculty of Mathematics, National Research University \u201cHigher School of Economics\u201d, 109028 Moscow, Russia"},{"name":"Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, 630090 Novosibirsk, Russia"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"663","DOI":"10.1006\/eujc.1999.0314","article-title":"Virtual knot theory","volume":"20","author":"Kauffman","year":"1999","journal-title":"Eur. 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