{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T15:40:22Z","timestamp":1772466022082,"version":"3.50.1"},"reference-count":50,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2026,2,20]],"date-time":"2026-02-20T00:00:00Z","timestamp":1771545600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,2,20]],"date-time":"2026-02-20T00:00:00Z","timestamp":1771545600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2026,3]]},"DOI":"10.1007\/s00454-025-00788-5","type":"journal-article","created":{"date-parts":[[2026,2,20]],"date-time":"2026-02-20T18:43:58Z","timestamp":1771613038000},"page":"294-319","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Hopf Arborescent Links, Minor Theory, and Decidability of the Genus Defect"],"prefix":"10.1007","volume":"75","author":[{"given":"Pierre","family":"Dehornoy","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-2060-3582","authenticated-orcid":false,"given":"Corentin","family":"Lunel","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Arnaud","family":"de Mesmay","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,2,20]]},"reference":[{"key":"788_CR1","doi-asserted-by":"crossref","unstructured":"The knot atlas: $$8_{10}$$. http:\/\/katlas.org\/wiki\/8_10. Accessed: 2024-13-03","DOI":"10.7146\/aprja.v13i1.151222"},{"issue":"9","key":"788_CR2","doi-asserted-by":"publisher","first-page":"3821","DOI":"10.1090\/S0002-9947-05-03919-X","volume":"358","author":"I Agol","year":"2006","unstructured":"Agol, I., Hass, J., Thurston, W.: The computational complexity of knot genus and spanning area. Trans. Am. Math. Soc. 358(9), 3821\u20133850 (2006)","journal-title":"Trans. Am. Math. Soc."},{"issue":"3","key":"788_CR3","doi-asserted-by":"publisher","first-page":"351","DOI":"10.4171\/lem\/59-3-8","volume":"59","author":"S Baader","year":"2014","unstructured":"Baader, S.: Positive braids of maximal signature. Enseign. Math. (2) 59(3), 351\u2013358 (2014)","journal-title":"Enseign. Math. (2)"},{"key":"788_CR4","unstructured":"Baader, S., Dehornoy, P.: Minor theory for surfaces and divides of maximal signature. arXiv preprint arXiv:1211.7348 (2012)"},{"key":"788_CR5","doi-asserted-by":"crossref","unstructured":"Baader, S., Dehornoy, P., Liechti, L.: Minor theory for quasipositive surfaces. In: Athanase Papadopoulos, editor, Essays in geometry, dedicated to Norbert A\u2019Campo, pages 351\u2013358. EMS Pres (2023)","DOI":"10.4171\/irma\/34\/17"},{"issue":"4","key":"788_CR6","doi-asserted-by":"publisher","first-page":"2639","DOI":"10.1090\/tran\/7051","volume":"370","author":"S Baader","year":"2018","unstructured":"Baader, S., Feller, P., Lewark, L., Liechti, L.: On the topological 4-genus of torus knots. Trans. Am. Math. Soc. 370(4), 2639\u20132656 (2018)","journal-title":"Trans. Am. Math. Soc."},{"key":"788_CR7","unstructured":"Baroni, F.: Classification of genus-two surfaces in $$\\mathbb{S}^{3}$$. arXiv preprint arXiv:2309.05387 (2023)"},{"key":"788_CR8","unstructured":"Bastl, S., Burke, R., Chatterjee, R., Dey, S., Durst, A., Friedl, S., Galvin, D., Rivas, A.G., Hirsch, T., Hobohm, C., Hsueh, C.-S., Kegel, M., Kern, F., Lee, S.M.S., L\u00f6h, C., Manikandan, N., Mousseau, L., Munser, L., Pencovitch, M., Perras, P., Powell, M., Quintanilha, J.P., Schambeck, L., Suchodoll, D., Tancer, M., Thiele, A., Tru\u00f6l, P., Uschold, M., Vesel\u00e1, S., Wei\u00df, M., von Wunsch-Rolshoven, M.: Algorithms in 4-manifold topology (2024). arXiv:2411.08775"},{"key":"788_CR9","unstructured":"Bonahon, F., Siebenmann, L.C.: New geometric splittings of classical knots and the classification and symmetries of arborescent knots. Preprint available on the authors\u2019 webpage (1979)"},{"key":"788_CR10","doi-asserted-by":"crossref","unstructured":"Burde, G., Zieschang, H.: Knots. Walter de gruyter (2002)","DOI":"10.1515\/9783110198034"},{"key":"788_CR11","unstructured":"Burton, B.A., Budney, R., Pettersson, W., et\u00a0al.: Regina: Software for low-dimensional topology. http:\/\/regina-normal.github.io\/ (1999)\u2013(2023)"},{"key":"788_CR12","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-21275-3","volume-title":"Parameterized algorithms","author":"M Cygan","year":"2015","unstructured":"Cygan, M., Fomin, F.V., Kowalik, \u0141, Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized algorithms, vol. 4. Springer, Berlin (2015)"},{"key":"788_CR13","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2021.107648","volume":"381","author":"A de Mesmay","year":"2021","unstructured":"de Mesmay, A., Rieck, Y., Sedgwick, E., Tancer, M.: The unbearable hardness of unknotting. Adv. Math. 381, 107648 (2021)","journal-title":"Adv. Math."},{"key":"788_CR14","volume-title":"Graph theory. Number 173 in Graduate texts in mathematics","author":"R Diestel","year":"2016","unstructured":"Diestel, R.: Graph theory. Number 173 in Graduate texts in mathematics, 5th edn. Springer, New York (2016)","edition":"5"},{"key":"788_CR15","unstructured":"Fox, R.: Some problems in knot theory. In: Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute), pages 168\u2013176, Englewood Cliffs, N.J. Prentice-Hall (1961)"},{"issue":"3","key":"788_CR16","doi-asserted-by":"publisher","first-page":"357","DOI":"10.4310\/jdg\/1214437136","volume":"17","author":"MH Freedman","year":"1982","unstructured":"Freedman, M.H.: The topology of four-dimensional manifolds. Journal of Differential Geometry 17(3), 357\u2013453 (1982)","journal-title":"Journal of Differential Geometry"},{"key":"788_CR17","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1090\/conm\/020\/718138","volume":"20","author":"D Gabai","year":"1983","unstructured":"Gabai, D.: The murasugi sum is a natural geometric operation. Contemp. Math. 20, 131\u2013143 (1983)","journal-title":"Contemp. Math."},{"key":"788_CR18","doi-asserted-by":"crossref","unstructured":"Gabai, D.: Genera of the Arborescent Links. Memoirs of the American Mathematical Society. American Mathematical Society (1986)","DOI":"10.1215\/S0012-7094-86-05336-6"},{"issue":"1","key":"788_CR19","doi-asserted-by":"publisher","first-page":"97","DOI":"10.2140\/gt.2006.10.97","volume":"10","author":"E Giroux","year":"2006","unstructured":"Giroux, E., Goodman, N.: On the stable equivalence of open books in three-manifolds. Geometry & Topology 10(1), 97\u2013114 (2006)","journal-title":"Geometry & Topology"},{"key":"788_CR20","doi-asserted-by":"crossref","unstructured":"McA, C., Gordon, Luecke, J.S.: Knots are determined by their complements. J. Am. Math. Soc., 2, 371\u2013415 (1989)","DOI":"10.1090\/S0894-0347-1989-0965210-7"},{"issue":"3","key":"788_CR21","doi-asserted-by":"publisher","first-page":"245","DOI":"10.1007\/BF02559591","volume":"105","author":"W Haken","year":"1961","unstructured":"Haken, W.: Theorie der normalfl\u00e4chen. Acta Math. 105(3), 245\u2013375 (1961)","journal-title":"Acta Math."},{"issue":"2","key":"788_CR22","doi-asserted-by":"publisher","first-page":"185","DOI":"10.1145\/301970.301971","volume":"46","author":"J Hass","year":"1999","unstructured":"Hass, J., Lagarias, J.C., Pippenger, N.: The computational complexity of knot and link problems. Journal of the ACM (JACM) 46(2), 185\u2013211 (1999)","journal-title":"Journal of the ACM (JACM)"},{"key":"788_CR23","volume-title":"Algebraic topology","author":"A Hatcher","year":"2002","unstructured":"Hatcher, A.: Algebraic topology. Cambridge University Press, Cambridge; New York (2002)"},{"issue":"6","key":"788_CR24","doi-asserted-by":"publisher","first-page":"797","DOI":"10.4310\/MRL.1994.v1.n6.a14","volume":"1","author":"PB Kronheimer","year":"1994","unstructured":"Kronheimer, P.B., Mrowka, T.S.: The genus of embedded surfaces in the projective plane. Mathematical Research Letters 1(6), 797\u2013808 (1994)","journal-title":"Mathematical Research Letters"},{"key":"788_CR25","first-page":"210","volume":"95","author":"JB Kruskal","year":"1960","unstructured":"Kruskal, J.B.: Well-quasi-ordering, the tree theorem, and vazsonyi\u2019s conjecture. Trans. Am. Math. Soc. 95, 210\u2013225 (1960)","journal-title":"Trans. Am. Math. Soc."},{"issue":"1","key":"788_CR26","doi-asserted-by":"publisher","first-page":"189","DOI":"10.2140\/pjm.2019.301.189","volume":"301","author":"G Kuperberg","year":"2019","unstructured":"Kuperberg, G.: Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization. Pac. J. Math. 301(1), 189\u2013241 (2019)","journal-title":"Pac. J. Math."},{"key":"788_CR27","doi-asserted-by":"crossref","unstructured":"Lackenby, M.: Elementary Knot Theory. In: Lectures on Geometry. Oxford University Press (01 2017)","DOI":"10.1093\/oso\/9780198784913.003.0002"},{"key":"788_CR28","doi-asserted-by":"publisher","first-page":"580","DOI":"10.1007\/s00454-017-9905-8","volume":"58","author":"M Lackenby","year":"2017","unstructured":"Lackenby, M.: Some conditionally hard problems on links and 3-manifolds. Discrete & Computational Geometry 58, 580\u2013595 (2017)","journal-title":"Discrete & Computational Geometry"},{"key":"788_CR29","doi-asserted-by":"crossref","unstructured":"Lackenby, M.: Algorithms in 3-manifold theory. Surveys in Differential Geometry (2020)","DOI":"10.4310\/SDG.2020.v25.n1.a5"},{"key":"788_CR30","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2021.107796","volume":"387","author":"M Lackenby","year":"2021","unstructured":"Lackenby, M.: The efficient certification of knottedness and thurston norm. Adv. Math. 387, 107796 (2021)","journal-title":"Adv. Math."},{"issue":"1","key":"788_CR31","doi-asserted-by":"publisher","first-page":"403","DOI":"10.2140\/agt.2020.20.403","volume":"20","author":"L Liechti","year":"2020","unstructured":"Liechti, L.: On the genus defect of positive braid knots. Algebraic & Geometric Topology 20(1), 403\u2013428 (2020)","journal-title":"Algebraic & Geometric Topology"},{"key":"788_CR32","unstructured":"Markov, A.A.: The insolubility of the problem of homeomorphy. In: Doklady Akademii Nauk, volume 121, pages 218\u2013220. Russian Academy of Sciences (1958)"},{"issue":"2","key":"788_CR33","doi-asserted-by":"publisher","first-page":"259","DOI":"10.4171\/jems\/252","volume":"13","author":"J Matou\u0161ek","year":"2010","unstructured":"Matou\u0161ek, J., Tancer, M., Wagner, U.: Hardness of embedding simplicial complexes in $$\\mathbb{r} ^d$$. J. Eur. Math. Soc. 13(2), 259\u2013295 (2010)","journal-title":"J. Eur. Math. Soc."},{"key":"788_CR34","volume-title":"Algorithmic topology and classification of 3-manifolds","author":"SV Matveev","year":"2007","unstructured":"Matveev, S.V.: Algorithmic topology and classification of 3-manifolds. Springer, Berlin (2007)"},{"key":"788_CR35","doi-asserted-by":"publisher","first-page":"575","DOI":"10.24033\/bsmf.2747","volume":"145","author":"F Misev","year":"2014","unstructured":"Misev, F.: Cutting arcs for torus links and trees. Bull. Soc. Math. France 145, 575\u2013602 (2014)","journal-title":"Bull. Soc. Math. France"},{"key":"788_CR36","unstructured":"Misev, F.: On the plumbing structure of fibre surfaces. PhD thesis, Universit\u00e4t Bern (2016)"},{"issue":"2","key":"788_CR37","first-page":"375","volume":"56","author":"F Misev","year":"2019","unstructured":"Misev, F.: Hopf bands in arborescent hopf plumbings. Osaka Journal of Mathematics 56(2), 375\u2013389 (2019)","journal-title":"Osaka Journal of Mathematics"},{"issue":"4","key":"788_CR38","doi-asserted-by":"publisher","first-page":"544","DOI":"10.2307\/2373107","volume":"85","author":"K Murasugi","year":"1963","unstructured":"Murasugi, K.: On a certain subgroup of the group of an alternating link. Am. J. Math. 85(4), 544\u2013550 (1963)","journal-title":"Am. J. Math."},{"issue":"4","key":"788_CR39","doi-asserted-by":"publisher","first-page":"833","DOI":"10.1017\/S0305004100003844","volume":"59","author":"CSJA Nash-Williams","year":"1963","unstructured":"Nash-Williams, C.S.J.A.: On well-quasi-ordering finite trees. Mathematical Proceedings of the Cambridge Philosophical Society 59(4), 833\u2013835 (1963)","journal-title":"Mathematical Proceedings of the Cambridge Philosophical Society"},{"issue":"1","key":"788_CR40","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1007\/s40598-015-0033-3","volume":"2","author":"B Ozbagci","year":"2015","unstructured":"Ozbagci, B., Popescu-Pampu, P.: Generalized plumbings and murasugi sums. Arnold Mathematical Journal 2(1), 69\u2013119 (2015)","journal-title":"Arnold Mathematical Journal"},{"issue":"2","key":"788_CR41","doi-asserted-by":"publisher","first-page":"581","DOI":"10.4007\/annals.2020.191.2.5","volume":"191","author":"L Piccirillo","year":"2020","unstructured":"Piccirillo, L.: The conway knot is not slice. Ann. Math. 191(2), 581\u2013591 (2020)","journal-title":"Ann. Math."},{"key":"788_CR42","doi-asserted-by":"crossref","unstructured":"Rolfsen, D.: Knots and links. AMS Chelsea Pub, Providence, R.I (2003). OCLC: ocm52901393","DOI":"10.1090\/chel\/346"},{"issue":"1","key":"788_CR43","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1090\/S0273-0979-1993-00397-5","volume":"29","author":"L Rudolph","year":"1993","unstructured":"Rudolph, L.: Quasipositivity as an obstruction to sliceness. Bull. Am. Math. Soc. 29(1), 51\u201359 (1993)","journal-title":"Bull. Am. Math. Soc."},{"key":"788_CR44","doi-asserted-by":"crossref","unstructured":"Rudolph, L.: Positive links are strongly quasipositive. Geometry & Topology Monographs, Volume 2: Proceedings of the Kirbyfest, pages 555\u2013562 (1998)","DOI":"10.2140\/gtm.1999.2.555"},{"key":"788_CR45","first-page":"861","volume":"31","author":"M Sakuma","year":"1994","unstructured":"Sakuma, M.: Minimal genus seifert surfaces for special arborescent links. Osaka Journal of Mathematics 31, 861\u2013905 (1994)","journal-title":"Osaka Journal of Mathematics"},{"key":"788_CR46","first-page":"315","volume":"27","author":"JR Stallings","year":"1975","unstructured":"Stallings, J.R.: Constructions of fibred knots and links. Proc. Symp. Pure Math., AMS 27, 315\u2013319 (1975)","journal-title":"Proc. Symp. Pure Math., AMS"},{"key":"788_CR47","unstructured":"Teichner, P.: Slice knots: Knot theory in the 4th dimension, 2011. Lecture notes by Julia Collins and Mark Powell. Electronic version available from https:\/\/www.maths.ed.ac.uk\/~v1ranick\/papers\/sliceknots2.pdf"},{"key":"788_CR48","doi-asserted-by":"publisher","first-page":"56","DOI":"10.2307\/1970594","volume":"87","author":"F Waldhausen","year":"1968","unstructured":"Waldhausen, F.: On irreducible 3-manifolds which are sufficiently large. Ann. Math. 87, 56\u201388 (1968)","journal-title":"Ann. Math."},{"key":"788_CR49","doi-asserted-by":"crossref","unstructured":"Weinberger, S.: Homology manifolds. Handbook of geometric topology, 1085\u20131102 (2002)","DOI":"10.1016\/B978-044482432-5\/50022-6"},{"key":"788_CR50","doi-asserted-by":"publisher","first-page":"373","DOI":"10.1016\/0040-9383(73)90029-3","volume":"12","author":"W Whitten","year":"1973","unstructured":"Whitten, W.: Isotopy types of knot spanning surface. Topology 12, 373\u2013380 (1973)","journal-title":"Topology"}],"container-title":["Discrete &amp; Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-025-00788-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00454-025-00788-5","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-025-00788-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T14:49:48Z","timestamp":1772462988000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00454-025-00788-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,20]]},"references-count":50,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["788"],"URL":"https:\/\/doi.org\/10.1007\/s00454-025-00788-5","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"value":"0179-5376","type":"print"},{"value":"1432-0444","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,20]]},"assertion":[{"value":"8 July 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 September 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 September 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 February 2026","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no relevant financial or non-financial interests to disclose.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}]}}