{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T13:51:21Z","timestamp":1768744281554,"version":"3.49.0"},"reference-count":15,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2019,12,1]],"date-time":"2019-12-01T00:00:00Z","timestamp":1575158400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2019,12,1]],"date-time":"2019-12-01T00:00:00Z","timestamp":1575158400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1122374"],"award-info":[{"award-number":["1122374"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2020,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A flip is a minimal move between two triangulations of a polytope. The set of triangulations of a polytope was shown by Santos to not always be connected by flips, and it is an interesting problem to find large classes of polytopes for which it is. One such class which has received considerable attention is the product of two simplices. Santos proved that the set of triangulations of a product of two simplices is connected by flips when one of the simplices is a triangle. However, the author showed that it is not connected when one of the simplices is four-dimensional and the other has very large dimension. In this paper we show that it is connected when one of the simplices is a tetrahedron, thereby extending Santos\u2019s result as far as possible.<\/jats:p>","DOI":"10.1007\/s00454-019-00157-z","type":"journal-article","created":{"date-parts":[[2019,12,2]],"date-time":"2019-12-02T16:10:36Z","timestamp":1575303036000},"page":"1-30","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Flip-Connectivity of Triangulations of the Product of a Tetrahedron and Simplex"],"prefix":"10.1007","volume":"63","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2212-9919","authenticated-orcid":false,"given":"Gaku","family":"Liu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2019,12,1]]},"reference":[{"issue":"2","key":"157_CR1","doi-asserted-by":"publisher","first-page":"495","DOI":"10.1016\/j.aim.2007.02.014","volume":"214","author":"F Ardila","year":"2007","unstructured":"Ardila, F., Billey, S.: Flag arrangements and triangulations of products of simplices. Adv. Math. 214(2), 495\u2013524 (2007)","journal-title":"Adv. Math."},{"issue":"3","key":"157_CR2","doi-asserted-by":"publisher","first-page":"485","DOI":"10.1007\/s00454-013-9485-1","volume":"49","author":"F Ardila","year":"2013","unstructured":"Ardila, F., Ceballos, C.: Acyclic systems of permutations and fine mixed subdivisions of simplices. Discrete Comput. Geom. 49(3), 485\u2013510 (2013)","journal-title":"Discrete Comput. Geom."},{"issue":"4","key":"157_CR3","doi-asserted-by":"publisher","first-page":"795","DOI":"10.1007\/s00209-008-0400-z","volume":"262","author":"F Ardila","year":"2009","unstructured":"Ardila, F., Develin, M.: Tropical hyperplane arrangements and oriented matroids. Math. Z. 262(4), 795\u2013816 (2009)","journal-title":"Math. Z."},{"key":"157_CR4","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-12971-1","volume-title":"Triangulations: Structures for Algorithms and Applications. Algorithms and Computation in Mathematics","author":"JA De Loera","year":"2010","unstructured":"De Loera, J.A., Rambau, J., Santos, F.: Triangulations: Structures for Algorithms and Applications. Algorithms and Computation in Mathematics, vol. 25. Springer, Berlin (2010)"},{"key":"157_CR5","doi-asserted-by":"crossref","first-page":"1","DOI":"10.4171\/dm\/154","volume":"9","author":"M Develin","year":"2004","unstructured":"Develin, M., Sturmfels, B.: Tropical convexity. Doc. Math. 9, 1\u201327 (2004)","journal-title":"Doc. Math."},{"key":"157_CR6","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-8176-4771-1","volume-title":"Discriminants, Resultants and Multidimensional Determinants. Mathematics: Theory & Applications.","author":"IM Gel\u2019fand","year":"1994","unstructured":"Gel\u2019fand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, Resultants and Multidimensional Determinants. Mathematics: Theory & Applications. Birkh\u00e4user, Boston (1994)"},{"key":"157_CR7","doi-asserted-by":"crossref","unstructured":"Horn, S.: A topological representation theorem for tropical oriented matroids. In: Proceedings of the 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012). Discrete Mathematics & Theoretical Computer Science Proceedings, AR, pp. 135\u2013146. DMTCS, Nancy (2012)","DOI":"10.46298\/dmtcs.3026"},{"key":"157_CR8","unstructured":"Liu, G.: Flip-connectivity of triangulations of the product of a tetrahedron and simplex (2016). arXiv:1601.06031"},{"issue":"4","key":"157_CR9","doi-asserted-by":"publisher","first-page":"810","DOI":"10.1007\/s00454-018-9971-6","volume":"59","author":"G Liu","year":"2018","unstructured":"Liu, G.: A zonotope and a product of two simplices with disconnected flip graphs. Discrete Comput. Geom. 59(4), 810\u2013842 (2018)","journal-title":"Discrete Comput. Geom."},{"issue":"2","key":"157_CR10","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1007\/s00454-001-0062-7","volume":"27","author":"D Maclagan","year":"2002","unstructured":"Maclagan, D., Thomas, R.R.: Combinatorics of the toric Hilbert scheme. Discrete Comput. Geom. 27(2), 249\u2013272 (2002)","journal-title":"Discrete Comput. Geom."},{"key":"157_CR11","doi-asserted-by":"crossref","unstructured":"Oh, S., Yoo, H.: Triangulations of $$\\Delta _{n-1} \\times \\Delta _{d-1}$$ and tropical oriented matroids. In: Proceedings of the 23th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011). Discrete Mathematics & Theoretical Computer Science Proceedings, AR, pp. 717\u2013728. DMTCS, Nancy (2011)","DOI":"10.46298\/dmtcs.2947"},{"issue":"3","key":"157_CR12","doi-asserted-by":"publisher","first-page":"611","DOI":"10.1090\/S0894-0347-00-00330-1","volume":"13","author":"F Santos","year":"2000","unstructured":"Santos, F.: A point set whose space of triangulations is disconnected. J. Am. Math. Soc. 13(3), 611\u2013637 (2000)","journal-title":"J. Am. Math. Soc."},{"issue":"741","key":"157_CR13","doi-asserted-by":"publisher","first-page":"0","DOI":"10.1090\/memo\/0741","volume":"156","author":"Francisco Santos","year":"2002","unstructured":"Santos, F.: Triangulations of Oriented Matroids. Memoirs of the American Mathematical Society, vol. 156(741). American Mathematical Society, Providence (2002)","journal-title":"Memoirs of the American Mathematical Society"},{"key":"157_CR14","series-title":"Contemporary Mathematics","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1090\/conm\/374\/06904","volume-title":"Integer Points in Polyhedra-Geometry, Number Theory, Algebra, Optimization","author":"F Santos","year":"2005","unstructured":"Santos, F.: The Cayley trick and triangulations of products of simplices. In: Barvinok, A., et al. (eds.) Integer Points in Polyhedra-Geometry, Number Theory, Algebra, Optimization. Contemporary Mathematics, vol. 374, pp. 151\u2013177. American Mathematical Society, Providence (2005)"},{"key":"157_CR15","doi-asserted-by":"crossref","unstructured":"Sturmfels, B.: Gr\u00f6bner Bases and Convex Polytopes. University Lecture Series, vol. 8. American Mathematical Society, Providence (1996)","DOI":"10.1090\/ulect\/008"}],"container-title":["Discrete &amp; Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-019-00157-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00454-019-00157-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-019-00157-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,23]],"date-time":"2023-09-23T14:56:45Z","timestamp":1695481005000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00454-019-00157-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,12,1]]},"references-count":15,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2020,1]]}},"alternative-id":["157"],"URL":"https:\/\/doi.org\/10.1007\/s00454-019-00157-z","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"value":"0179-5376","type":"print"},{"value":"1432-0444","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,12,1]]},"assertion":[{"value":"2 April 2016","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 June 2019","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 November 2019","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 December 2019","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}