{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,26]],"date-time":"2025-11-26T04:38:07Z","timestamp":1764131887051,"version":"3.45.0"},"reference-count":30,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,6,27]],"date-time":"2025-06-27T00:00:00Z","timestamp":1750982400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,6,27]],"date-time":"2025-06-27T00:00:00Z","timestamp":1750982400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001774","name":"University of Sydney","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100001774","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Appl. and Comput. Topology"],"published-print":{"date-parts":[[2025,9]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>We present a practical algorithm to test whether a 3-manifold given by a triangulation or an ideal triangulation contains a closed essential surface. This property has important theoretical and algorithmic consequences. As a testament to its practicality, we run the algorithm over a comprehensive body of closed 3-manifolds and knot exteriors, yielding results that were not previously known. The algorithm derives from the original Jaco\u2013Oertel framework, involves both enumeration and optimisation procedures, and combines several techniques from normal surface theory. Our methods are relevant for other difficult computational problems in 3-manifold theory, such as the recognition problem for knots, links and 3-manifolds.<\/jats:p>","DOI":"10.1007\/s41468-025-00208-w","type":"journal-article","created":{"date-parts":[[2025,6,26]],"date-time":"2025-06-26T23:26:51Z","timestamp":1750980411000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Computing closed essential surfaces in 3-manifolds"],"prefix":"10.1007","volume":"9","author":[{"given":"Benjamin A.","family":"Burton","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6731-0327","authenticated-orcid":false,"given":"Stephan","family":"Tillmann","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,6,27]]},"reference":[{"key":"208_CR1","unstructured":"Bell, M.: Flipper (computer software). pypi.python.org\/pypi\/flipper (2013\u20132017)"},{"key":"208_CR2","doi-asserted-by":"crossref","unstructured":"Burton, B.A.: Computational topology with Regina: algorithms, heuristics and implementations. In: Geometry and Topology Down Under, Contemporary Mathematics, vol. 597, pp. 195\u2013224. American Mathematical Society, Providence (2013)","DOI":"10.1090\/conm\/597\/11877"},{"issue":"1","key":"208_CR3","doi-asserted-by":"publisher","first-page":"116","DOI":"10.1007\/s00454-014-9572-y","volume":"52","author":"BA Burton","year":"2014","unstructured":"Burton, B.A.: A new approach to crushing 3-manifold triangulations. Discrete Comput. Geom. 52(1), 116\u2013139 (2014)","journal-title":"Discrete Comput. Geom."},{"key":"208_CR4","unstructured":"Burton, B.A., Ozlen M.: A fast branching algorithm for unknot recognition with experimental polynomial-time behaviour. Mathematical Programming, to appear, arXiv:1211.1079 (2012)"},{"issue":"4","key":"208_CR5","doi-asserted-by":"publisher","first-page":"772","DOI":"10.1007\/s00453-012-9645-3","volume":"65","author":"BA Burton","year":"2013","unstructured":"Burton, B.A., Ozlen, M.: A tree traversal algorithm for decision problems in knot theory and 3-manifold topology. Algorithmica 65(4), 772\u2013801 (2013)","journal-title":"Algorithmica"},{"key":"208_CR6","unstructured":"Burton, B.A., Budney R., Pettersson W., et\u00a0al.: Regina: Software for 3-manifold topology and normal surface theory. http:\/\/regina.sourceforge.net\/ (1999\u20132012)"},{"key":"208_CR7","doi-asserted-by":"crossref","unstructured":"Burton, B.A., Rubinstein, J.H., Tillmann, S.: The Weber-Seifert dodecahedral space is non-Haken. Trans. Am. Math. Soc. 364(2), 911\u2013932 (2012)","DOI":"10.1090\/S0002-9947-2011-05419-X"},{"key":"208_CR8","doi-asserted-by":"crossref","unstructured":"Burton, B.A., Coward, A., Tillmann, S.: Computing closed essential surfaces in knot complements. In: Computational Geometry (SoCG\u201913). ACM, New York, pp.\u00a0405\u2013413 (2013)","DOI":"10.1145\/2493132.2462380"},{"issue":"2","key":"208_CR9","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1016\/S0166-8641(98)00043-1","volume":"96","author":"E Finkelstein","year":"1999","unstructured":"Finkelstein, E., Moriah, Y.: Tubed incompressible surfaces in knot and link complements. Topol. Appl. 96(2), 153\u2013170 (1999)","journal-title":"Topol. Appl."},{"issue":"6","key":"208_CR10","doi-asserted-by":"publisher","first-page":"1068","DOI":"10.1137\/0220067","volume":"20","author":"JL Hafner","year":"1991","unstructured":"Hafner, J.L., McCurley, K.S.: Asymptotically fast triangularization of matrices over rings. SIAM J. Comput. 20(6), 1068\u20131083 (1991)","journal-title":"SIAM J. Comput."},{"key":"208_CR11","unstructured":"Haken W.: Some results on surfaces in 3-manifolds. In: Studies in Modern Topology, Studies in Mathematics, vol. 5, pp.\u00a039\u201398. Mathematical Association of America (1968)"},{"key":"208_CR12","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, 245\u2013375 (1961)","journal-title":"Acta Math."},{"issue":"1","key":"208_CR13","first-page":"125","volume":"11","author":"RC Haraway III","year":"2020","unstructured":"Haraway, R.C., III.: Determining hyperbolicity of compact orientable 3-manifolds with torus boundary. J. Comput. Geom. 11(1), 125\u2013136 (2020)","journal-title":"J. Comput. Geom."},{"issue":"4","key":"208_CR14","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1007\/BF03025227","volume":"20","author":"J Hoste","year":"1998","unstructured":"Hoste, J., Thistlethwaite, M., Weeks, J.: The first 1,701,936 knots. Math. Intell. 20(4), 33\u201348 (1998)","journal-title":"Math. Intell."},{"key":"208_CR15","doi-asserted-by":"crossref","unstructured":"Jaco, W.H., Shalen, P.B.: Seifert fibered spaces in $$3$$-manifolds. Mem. Am. Math. Soc. 21(220): viii+192 (1979)","DOI":"10.1090\/memo\/0220"},{"issue":"2","key":"208_CR16","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1016\/0040-9383(84)90039-9","volume":"23","author":"W Jaco","year":"1984","unstructured":"Jaco, W., Oertel, U.: An algorithm to decide if a $$3$$-manifold is a Haken manifold. Topology 23(2), 195\u2013209 (1984)","journal-title":"Topology"},{"issue":"1","key":"208_CR17","doi-asserted-by":"publisher","first-page":"61","DOI":"10.4310\/jdg\/1090503053","volume":"65","author":"W Jaco","year":"2003","unstructured":"Jaco, W., Rubinstein, J.H.: 0-efficient triangulations of 3-madnifolds. J. Differ. Geom. 65(1), 61\u2013168 (2003)","journal-title":"J. Differ. Geom."},{"issue":"3","key":"208_CR18","first-page":"358","volume":"39","author":"W Jaco","year":"1995","unstructured":"Jaco, W., Tollefson, J.L.: Algorithms for the complete decomposition of a closed $$3$$-manifold. Ill. J. Math. 39(3), 358\u2013406 (1995)","journal-title":"Ill. J. Math."},{"key":"208_CR19","doi-asserted-by":"crossref","unstructured":"Johannson, K.: Homotopy Equivalences of $$3$$-Manifolds with Boundaries. Lecture Notes in Mathematics, vol. 761. Springer, Berlin (1979)","DOI":"10.1007\/BFb0085406"},{"key":"208_CR20","doi-asserted-by":"crossref","unstructured":"Johannson, K.: On the mapping class group of simple $$3$$-manifolds. In: Topology of Low-Dimensional Manifolds (Proceedings of the Second Sussex Conference, Chelwood Gate, Lecture Notes in Mathematics, vol. 722, pp. 48\u201366. Springer, Berlin 1979 (1977)","DOI":"10.1007\/BFb0063189"},{"issue":"3","key":"208_CR21","doi-asserted-by":"publisher","first-page":"305","DOI":"10.1017\/S1446788700008557","volume":"78","author":"E Kang","year":"2005","unstructured":"Kang, E.: Normal surfaces in non-compact 3-manifolds. J. Aust. Math. Soc. 78(3), 305\u2013321 (2005)","journal-title":"J. Aust. Math. Soc."},{"issue":"4","key":"208_CR22","doi-asserted-by":"publisher","first-page":"499","DOI":"10.1137\/0208040","volume":"8","author":"R Kannan","year":"1979","unstructured":"Kannan, R., Bachem, A.: Polynomial algorithms for computing the Smith and Hermite normal forms of an integer matrix. SIAM J. Comput. 8(4), 499\u2013507 (1979)","journal-title":"SIAM J. Comput."},{"key":"208_CR23","series-title":"Algorithms and Computation in Mathematics","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-05102-3","volume-title":"Algorithmic Topology and Classification of 3-Manifolds","author":"S Matveev","year":"2003","unstructured":"Matveev, S.: Algorithmic Topology and Classification of 3-Manifolds. Algorithms and Computation in Mathematics, vol. 9. Springer, Berlin (2003)"},{"issue":"1","key":"208_CR24","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1016\/0040-9383(84)90023-5","volume":"23","author":"W Menasco","year":"1984","unstructured":"Menasco, W.: Closed incompressible surfaces in alternating knot and link complements. Topology 23(1), 37\u201344 (1984)","journal-title":"Topology"},{"issue":"1","key":"208_CR25","doi-asserted-by":"publisher","first-page":"209","DOI":"10.2140\/pjm.1984.111.209","volume":"111","author":"U Oertel","year":"1984","unstructured":"Oertel, U.: Closed incompressible surfaces in complements of star links. Pac. J. Math. 111(1), 209\u2013230 (1984)","journal-title":"Pac. J. Math."},{"key":"208_CR26","first-page":"955","volume-title":"Handbook of Geometric Topology","author":"PB Shalen","year":"2002","unstructured":"Shalen, P.B.: Representations of 3-manifold groups. In: Sher, R.B., Daverman, R.J. (eds.) Handbook of Geometric Topology, pp. 955\u20131044. Elsevier, Amsterdam (2002)"},{"issue":"2","key":"208_CR27","doi-asserted-by":"publisher","first-page":"203","DOI":"10.2307\/1971277","volume":"124","author":"WP Thurston","year":"1986","unstructured":"Thurston, W.P.: Hyperbolic structures on $$3$$-manifolds. I. Deformation of acylindrical manifolds. Ann. Math. 124(2), 203\u2013246 (1986)","journal-title":"Ann. Math."},{"issue":"2","key":"208_CR28","first-page":"329","volume":"54","author":"S Tillmann","year":"2008","unstructured":"Tillmann, S.: Normal surfaces in topologically finite 3-manifolds. Enseign. Math. 54(2), 329\u2013380 (2008)","journal-title":"Enseign. Math."},{"issue":"2","key":"208_CR29","doi-asserted-by":"publisher","first-page":"359","DOI":"10.2140\/pjm.1998.183.359","volume":"183","author":"JL Tollefson","year":"1998","unstructured":"Tollefson, J.L.: Normal surface $$Q$$-theory. Pac. J. Math. 183(2), 359\u2013374 (1998)","journal-title":"Pac. J. Math."},{"issue":"2","key":"208_CR30","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(2), 56\u201388 (1968)","journal-title":"Ann. Math."}],"container-title":["Journal of Applied and Computational Topology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s41468-025-00208-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s41468-025-00208-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s41468-025-00208-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,12]],"date-time":"2025-11-12T11:25:15Z","timestamp":1762946715000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s41468-025-00208-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,27]]},"references-count":30,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,9]]}},"alternative-id":["208"],"URL":"https:\/\/doi.org\/10.1007\/s41468-025-00208-w","relation":{},"ISSN":["2367-1726","2367-1734"],"issn-type":[{"type":"print","value":"2367-1726"},{"type":"electronic","value":"2367-1734"}],"subject":[],"published":{"date-parts":[[2025,6,27]]},"assertion":[{"value":"5 January 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 August 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 February 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 June 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 November 2025","order":6,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Update","order":7,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Some of the details in references and citations were incorrect; It has been corrected now.","order":8,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"On behalf of all authors, the corresponding author states that there is no Conflict of interest","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"18"}}