부산대학교 도서관

The central question under investigation in this paper is how totranslate the idea of voltage operations, which have an extensivehistory in the theory of configurations [T. Pisanski and B.Servatius, {\it Configurations from a graphical viewpoint},Birkhäuser Adv. Texts Basler Lehrbücher, Birkhäuser/Springer, NewYork, 2013; MR2978043; M. Boben and T. Pisanski,European J. Combin. {\bf 24} (2003), no.~4, 431--457; MR1975946], maps on surfaces [A. Orbanić et al., ArsMath. Contemp. {\bf 4} (2011), no.~2, 385--402; MR2980587; A. Orbanić, D. Pellicer and A.~I. Weiss, J.Combin. Theory Ser. A {\bf 117} (2010), no.~4, 411--429; MR2592891; T. Pisanski, G. Williams and L.~W. Berman, Symmetry{\bf 9} (2017), no. 11, Paper No. 274, \doi{10.3390/sym9110274}; H.Koike et al., Electron. J. Combin. {\bf 24} (2017), no.~1, Paper No.1.3; MR3609173], and symmetric graphs [E.~T.Dobson, A. Malnič and D. Marušič, {\it Symmetry in graphs},Cambridge Stud. Adv. Math., 198, Cambridge Univ. Press, Cambridge,2022; MR4404766], into the setting of abstractpolytopes and maniplexes. Voltage graphs have previously been used tosolve problems in the study of abstract polytopes [e.g., J.~E.~M.Quesnel, {\it Abstract polytopes from their symmetry type graphs},Ph.D. thesis, Univ. Nacional Autómata México, 2021], but thispaper's aim is to develop a more complete and coherent theory of how todescribe, define, and apply voltage operations to polytopes and relatedobjects, as well as to better understand what properties are preservedin the resulting object. In many ways this paper is an attempt toaddress the same problems that motivated the development of stratifiedoperations in [G. Cunningham, D. Pellicer and G.~I. Williams,Algebr. Comb. {\bf 5} (2022), no.~2, 267--287; MR4426439], but it provides a more general and alternateframework for doing so, and provides a ready mechanism for doing sowhere the goal is to be able to describe the connection (monodromy)group of the polytope (or maniplex, or premaniplex), or to construct apolytope with a particular symmetry group. An advantage of the voltageoperation approach to the study of polytopes obtained from operationsapplied to symmetric polytopes (or maniplexes and premaniplexes) isthat they readily admit algorithmic implementations.\par Associated with each voltage operator is a premaniplex $\Cal Y$, and avoltage assignment $\eta$ that (essentially) associates to the edges of$\Cal Y$ elements of the connection group of the universal rank-$n$maniplex $U^{n}$, which is the flag graph of the universal polytope$\{\infty,\infty,\dots,\infty\}$ (for a detailed treatment of theuniversal polytope see [P. McMullen and E. Schulte, {\it Abstractregular polytopes}, Encyclopedia Math. Appl., 92, Cambridge Univ.Press, Cambridge, 2002 (§3D); MR1965665]). Conditionson the premaniplex $\Cal Y$ and the assignment $\eta$ are provided forwhen the associated operator preserves connectivity and coveringrelations. Details and restrictions on composition of voltage operatorsare also explored.\par This paper presents details of a range of classical polytopeoperations that can be interpreted and implemented concretely asvoltage operations on premaniplexes, maniplexes and abstract polytopes.These include the mix operator, snub, prism, pyramid, trapezotope, and$k$-bubble, as well as a description of how to implement the $\CalM\mapsto \widehat 2^{\Cal M}$ as a voltage operation, though in thislatter case the premaniplex $\Cal Y$ depends on the regular premaniplex$\Cal M$.\par A number of open problems are discussed, including the determinationof necessary and sufficient conditions on the premaniplex $\Cal Y$ andthe voltage assignment $\eta$ such that the associated rooted voltageoperation preserves polytopality of maniplexes.