부산대학교 도서관

The central question under investigation in this paper is how to translate the idea of voltage operations, which have an extensive history in the theory of configurations [T. Pisanski and B. Servatius, {\it Configurations from a graphical viewpoint}, Birkhäuser Adv. Texts Basler Lehrbücher, Birkhäuser/Springer, New York, 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., Ars Math. 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 abstract polytopes and maniplexes. Voltage graphs have previously been used to solve 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 this paper's aim is to develop a more complete and coherent theory of how to describe, define, and apply voltage operations to polytopes and related objects, as well as to better understand what properties are preserved in the resulting object. In many ways this paper is an attempt to address the same problems that motivated the development of stratified operations 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 alternate framework for doing so, and provides a ready mechanism for doing so where the goal is to be able to describe the connection (monodromy) group of the polytope (or maniplex, or premaniplex), or to construct a polytope with a particular symmetry group. An advantage of the voltage operation approach to the study of polytopes obtained from operations applied to symmetric polytopes (or maniplexes and premaniplexes) is that they readily admit algorithmic implementations. \par Associated with each voltage operator is a premaniplex $\Cal Y$, and a voltage 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 the universal polytope see [P. McMullen and E. Schulte, {\it Abstract regular polytopes}, Encyclopedia Math. Appl., 92, Cambridge Univ. Press, Cambridge, 2002 (§3D); MR1965665]). Conditions on the premaniplex $\Cal Y$ and the assignment $\eta$ are provided for when the associated operator preserves connectivity and covering relations. Details and restrictions on composition of voltage operators are also explored. \par This paper presents details of a range of classical polytope operations that can be interpreted and implemented concretely as voltage 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 $\Cal M\mapsto \widehat 2^{\Cal M}$ as a voltage operation, though in this latter case the premaniplex $\Cal Y$ depends on the regular premaniplex $\Cal M$. \par A number of open problems are discussed, including the determination of necessary and sufficient conditions on the premaniplex $\Cal Y$ and the voltage assignment $\eta$ such that the associated rooted voltage operation preserves polytopality of maniplexes.