{"pk":64950,"title":"The merging operation and \\((d-i)\\)-simplicial \\(i\\)-simple \\(d\\)-polytopes","subtitle":null,"abstract":"We define a certain merging operation that given two \\(d\\)-polytopes \\(P\\) and \\(Q\\) such that \\(P\\) has a simplex facet and \\(Q\\) has a simple vertex produces a new \\(d\\)-polytope \\(P \\triangleright Q\\) with \\(f_0(P)+f_0(Q)-(d+1)\\) vertices. We show that if for some \\(1\\leq i\\leq d-1\\), \\(P\\) and \\(Q\\) are \\((d-i)\\)-simplicial \\(i\\)-simple \\(d\\)-polytopes, then so is \\(P \\triangleright Q\\). We then use this operation to construct new families of \\((d-i)\\)-simplicial \\(i\\)-simple \\(d\\)-polytopes. Specifically, we prove that for all \\(2\\leq i \\leq d-2\\leq 6\\) with the exception of \\((i,d)=(3,8)\\) and \\((5,8)\\), there is an infinite family of \\((d-i)\\)-simplicial \\(i\\)-simple \\(d\\)-polytopes; furthermore, for all \\(2\\leq i\\leq 4\\), there is an infinite family of self-dual \\(i\\)-simplicial \\(i\\)-simple \\(2i\\)-polytopes. Finally, we show that for every \\(d\\geq 4\\), there are \\(2^{\\Omega(N)}\\) combinatorial types of \\((d-2)\\)-simplicial \\(2\\)-simple \\(d\\)-polytopes with at most \\(N\\) vertices.\n \nMathematics Subject Classifications: 52B05, 52B11\n \nKeywords: Connected sums, face lattice, face numbers, Gosset-Elte polytopes, self-dual polytopes","language":"en","license":{"name":"Creative Commons Attribution 4.0","short_name":"CC BY 4.0","text":"Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.","url":"https://creativecommons.org/licenses/by/4.0"},"keywords":[{"word":"Connected sums"},{"word":"face lattice"},{"word":"face numbers"},{"word":"Gosset-Elte polytopes"},{"word":"self-dual polytopes"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/3sf817pd","frozenauthors":[{"first_name":"Isabella","middle_name":"","last_name":"Novik","name_suffix":"","institution":"Department of Mathematics, University of Washington, Seattle, WA 98195-4350, U.S.A.","department":""},{"first_name":"Hailun","middle_name":"","last_name":"Zheng","name_suffix":"","institution":"Department of Mathematics, University of Houston-Downtown, One Main Street, Houston, TX 77002, U.S.A.","department":""}],"date_submitted":"2024-09-26T13:58:21Z","date_accepted":"2024-09-26T13:58:21Z","date_published":"2024-09-30T07:00:00Z","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/64950/galley/49760/download/"}]}