{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,18]],"date-time":"2026-08-18T05:09:30Z","timestamp":1787029770914,"version":"3.56.0"},"reference-count":19,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,8,18]],"date-time":"2022-08-18T00:00:00Z","timestamp":1660780800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Dorothea Schl\u00f6zer Postdoctoral Program by Georg-August University, G\u00f6ttingen"},{"name":"Slovenian Research Agency program P1-0292"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we study knotoids with extra graphical structure (bonded knotoids) in the settings of rigid vertex and topological vertex graphs. We construct bonded knotoid invariants by applying tangle insertion and unplugging at bonding sites of a bonded knotoid. We demonstrate that our invariants can be used for the analysis of the topological structure of proteins.<\/jats:p>","DOI":"10.3390\/sym14081724","type":"journal-article","created":{"date-parts":[[2022,8,19]],"date-time":"2022-08-19T02:48:23Z","timestamp":1660877303000},"page":"1724","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Invariants of Bonded Knotoids and Applications to Protein Folding"],"prefix":"10.3390","volume":"14","author":[{"given":"Neslihan","family":"G\u00fcg\u00fcmc\u00fc","sequence":"first","affiliation":[{"name":"Department of Mathematics, Izmir Institute of Technology, G\u00fclbah\u00e7e Campus, Izmir 35430, Turkey"},{"name":"Mathematics Institute, Georg-August University G\u00f6ttingen, Bunsenstrasse 3-5, 37073 G\u00f6ttingen, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8272-5392","authenticated-orcid":false,"given":"Bostjan","family":"Gabrovsek","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Louis H.","family":"Kauffman","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 South Morgan St., Chicago, IL 60607-7045, USA"},{"name":"Department of Mechanics and Mathematics, Novosibirsk State University, Novosibirsk 630090, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,18]]},"reference":[{"key":"ref_1","unstructured":"Dabrowski-Tumanski, P., Goundaroulis, D., Stasiak, A., and Sulkowska, J.I. (2019). \u03b8-curves in proteins. arXiv."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"9181","DOI":"10.1093\/nar\/gky559","article-title":"Two convergent pathways of DNA knotting in replicating DNA molecules as revealed by \u0398-curve analysis","volume":"46","author":"Stasiak","year":"2018","journal-title":"Nucleic Acids Res."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"586","DOI":"10.1111\/sapm.12357","article-title":"An invariant for colored bonded knots","volume":"146","year":"2021","journal-title":"Stud. Appl. Math."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Goundaroulis, D., G\u00fcg\u00fcmc\u00fc, N., Lambropoulou, S., Dorier, J., Stasiak, A., and Kauffman, L. (2017). Topological models for open-knotted protein chains using the concepts of knotoids and bonded knotoids. Polymers, 9.","DOI":"10.3390\/polym9090444"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Kauffman, L.H., and Magarshak, Y.B. (1995). Vassiliev Knot Invariants and the Structure of RNA Folding, World Scientific.","DOI":"10.1142\/9789812796189_0009"},{"key":"ref_6","unstructured":"Diamantis, I. (2021). Knotoids, pseudo knotoids, braidoids and pseudo braidoids on the torus. arXiv."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1016\/j.ejc.2017.06.004","article-title":"New invariants of knotoids","volume":"65","author":"Kauffman","year":"2017","journal-title":"Eur. J. Comb."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"6309","DOI":"10.1038\/s41598-017-06649-3","article-title":"Studies of global and local entanglements of individual protein chains using the concept of knotoids","volume":"7","author":"Goundaroulis","year":"2017","journal-title":"Sci. Rep."},{"key":"ref_9","first-page":"195","article-title":"Knotoids","volume":"49","author":"Turaev","year":"2012","journal-title":"Osaka J. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"697","DOI":"10.1090\/S0002-9947-1989-0946218-0","article-title":"Invariants of graphs in three-space","volume":"311","author":"Kauffman","year":"1989","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1515\/mlbmb-2017-0002","article-title":"A knot polynomial invariant for analysis of topology of RNA stems and Protein disulfide bonds","volume":"5","author":"Tian","year":"2017","journal-title":"Comput. Math. Biophys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"300","DOI":"10.1006\/aama.1996.0511","article-title":"Rational tangles","volume":"18","author":"Goldman","year":"1997","journal-title":"Adv. Appl. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/j.aam.2003.06.002","article-title":"On the classification of rational tangles","volume":"33","author":"Kauffman","year":"2004","journal-title":"Adv. Appl. Math."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Kauffman, L.H. (1988). New Invariants in the Theory of Knots, Soci\u00e9t\u00e9 Math\u00e9matique de France Ast\u00e9risque.","DOI":"10.2307\/2323625"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1340007","DOI":"10.1142\/S0218216513400075","article-title":"An affine index polynomial invariant of virtual knots","volume":"22","author":"Kauffman","year":"2013","journal-title":"J. Knot Theory Ramif."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Kauffman, L.H. (2005). Knot diagrammatics. Handbook of Knot Theory, Elsevier Science.","DOI":"10.1016\/B978-044451452-3\/50007-1"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Gabrov\u0161ek, B., and G\u00fcg\u00fcmc\u00fc, N. (2022). Invariants of multi-knotoids. arXiv.","DOI":"10.1007\/s00009-023-02370-w"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1940019","DOI":"10.1142\/S0218216519400194","article-title":"Rail knotoids","volume":"28","author":"Kodokostas","year":"2019","journal-title":"J. Knot Theory Ramif."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2150076","DOI":"10.1142\/S0218216521500760","article-title":"Parity, virtual closure and minimality of knotoids","volume":"30","author":"Kauffman","year":"2021","journal-title":"J. Knot Theory Ramif."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/8\/1724\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:11:32Z","timestamp":1760141492000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/8\/1724"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,18]]},"references-count":19,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2022,8]]}},"alternative-id":["sym14081724"],"URL":"https:\/\/doi.org\/10.3390\/sym14081724","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,8,18]]}}}