| Title: | Answers to questions about medial layer graphs of self-dual regular and chiral polytopes |
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| Authors: | ID Conder, Marston (Author)
ID Steinmann, Isabelle (Author) |
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| Files: |  AMC_Conder_Marston_Steinmann_Isabelle_2025.pdf (576,38 KB)
MD5: 453A8D5C2DDBB2914CF1FA5F72ECE26A
 https://amc-journal.eu/index.php/amc/article/view/3229
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| Language: | English |
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| Work type: | Article |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: | ZUP - University of Primorska Press
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| Abstract: | An abstract n-polytope P is a partially-ordered set which captures important properties of a geometric polytope, for any dimension n. For even n ≥ 2, the incidences between elements in the middle two layers of the Hasse diagram of P give rise to the medial layer graph of P, denoted by G = G(P). If n = 4, and P is both highly symmetric and self-dual of type {p, q, p}, then a Cayley graph C covering G can be constructed on a group of polarities of P. In this paper we address some open questions about the relationship between G and C that were raised in a 2008 paper by Monson and Weiss, and describe some interesting examples of these graphs. In particular, we give the first known examples of improperly self-dual chiral polytopes of type {3, q, 3}, which are also among the very few known examples of highly symmetric self-dual finite polytopes that do not admit a polarity. Also we show that if p = 3 then C cannot have a higher degree of s-arc-transitivity than G, and we present a family of regular 4-polytopes of type {6, q, 6} for which the vertex-stabilisers in the automorphism group of C are larger than those for G. |
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| Keywords: | abstract polytope, regular polytope, chiral polytope, medial graph |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 06.02.2025 |
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| Year of publishing: | 2025 |
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| Number of pages: | 18 |
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| Numbering: | Vol. 25, no. 1, [article no.] P1.01 |
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| PID: | 20.500.12556/RUP-21733  |
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| ISSN: | 1855-3966 |
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| UDC: | 519.17 |
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| eISSN: | 1855-3974 |
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| DOI: | 10.26493/1855-3974.3229.8b1 |
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| Publication date in RUP: | 16.09.2025 |
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| Views: | 409 |
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| Downloads: | 16 |
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