20.500.12556/RUP-21733 Answers to questions about medial layer graphs of self-dual regular and chiral polytopes Odgovori na vprašanja v zvezi z medialnimi slojnimi grafi samo-dualnih in kiralnih politopov 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. Abstrakten n-politop P je delno urejena množica, ki ima nekatere pomembne lastnosti geometrijskega politopa, za katerokoli dimenzijo n. Za sode n ≥ 2, incidence med elementi v srednjih dveh slojih Hassejevega diagrama politopa P določajo medialni slojni graf politopa P, označen z G = G(P). Če je n = 4 in je P visoko simetricen in samo-dualen tipa {p, q, p}, potem se Cayleyjev graf C, ki je krov nad G, da konstruirati na grupi polarnosti politopa P. V tem članku obravnavamo nekaj odprtih vprašanj v zvezi z odnosom med G in C, ki sta jih v članku iz leta 2008 zastavila Monson in Weiss, in opišemo nekaj zanimivih primerov teh grafov. Med drugim predstavimo prve znane primere nepravilno samo-dualnih kiralnih politopov tipa {3, q, 3}, ki so tudi med zelo redkimi znanimi primeri visoko simetricnih samo-dualnih končnih politopov, ki ne dopuščajo polarnosti. Pokažemo tudi, da ce je p = 3, potem C ne more imeti višje stopnje s-locne-tranzitivnosti kot G, in predstavimo družino regularnih 4-politopov tipa {6, q, 6}, za katere so stabilizatorji točk v grupi avtomorfizmov Cayleyjevega grafa C večji kot tisti od G. abstract polytope regular polytope chiral polytope medial graph abstraktni politop regularni politop kiralni politop medialni graf true false false Angleški jezik Slovenski jezik Članek v reviji 2025-09-13 23:23:55 2025-09-16 09:59:09 2026-02-02 13:20:24 0000-00-00 00:00:00 2025 0 0 18 no. 1, [article no.] P1.01 Vol. 25 2025 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2025-02-06 1855-3966 519.17 1855-3974 10.26493/1855-3974.3229.8b1 AMC_Conder_Marston_Steinmann_Isabelle_2025.pdf AMC_Conder_Marston_Steinmann_Isabelle_2025.pdf 1 453A8D5C2DDBB2914CF1FA5F72ECE26A 990a8e238841669d32983da91db4bd212ec65187b514433f5b7911cc3cc81f9a cdc06662-9223-11f0-8f0b-005056ac49c0 https://repozitorij.upr.si/Dokument.php?lang=slv&id=31607 https://amc-journal.eu/index.php/amc/article/view/3229 1 d7465487-9223-11f0-8f0b-005056ac49c0 https://repozitorij.upr.si/Dokument.php?lang=slv&id=31608 Založba Univerze na Primorskem 0 0 0