On the genera of polyhedral embeddings of cubic graph

In this article we present theoretical and computational results on the existence of polyhedral embeddings of graphs. The emphasis is on cubic graphs. We also describe an efficient algorithm to compute all polyhedral embeddings of a given cubic graph and constructions for cubic graphs with some special properties of their polyhedral embeddings. Some key results are that even cubic graphs with a polyhedral embedding on the torus can also have polyhedral embeddings in arbitrarily high genus, in fact in a genus {\em close} to the theoretical maximum for that number of vertices, and that there is no bound on the number of genera in which a cubic graph can have a polyhedral embedding. While these results suggest a large variety of polyhedral embeddings, computations for up to 28 vertices suggest that by far most of the cubic graphs do not have a polyhedral embedding in any genus and that the ratio of these graphs is increasing with the number of vertices.

Upper Frasnian Tornoceratidae (Ammonoidea) from the Sand Formation (Bergisch Gladbach-Paffrath Syncline, Rhenish Massif)

Excavations in the Sand district of Bergisch Gladbach (Rhenish Massif, Germany) yielded a rich ammonoid fauna of the upper Frasnian “Archoceras” varicosum Zone (Upper Devonian I-K, interval between the two Kellwasser levels). The previously unknown assemblages include six tornoceratid genera with 20 species, including seven new species (Aulatornoceras steinhauseni sp. nov., Aul. frenklerae sp. nov., Aul. ventrosulcatum sp. nov., Crassotornoceras nudum sp. nov., Cr. hetzeneggeri sp. nov., Retrotornoceras juxi sp. nov., Tornoceras aequilobum sp. nov.) and taxa described in open nomenclature. There are five associated gephuroceratid genera with nine species. The unexpectedly high genus- and species-level diversity at Sand, supported by statistical indices, is unprecedented compared to other contemporaneous ammonoid faunas. It highlights the currently fragmentary knowledge of top-Frasnian ammonoid faunas on a global scale. Phoenixites frechi, the dominant tornoceratid of hypoxic and organic-rich Kellwasser facies of Europe and North Africa, is completely missing at Sand. The local assemblage structure is analyzed statistically and interpreted in terms of palaeoecology. The occurrence of a new type of “Housean Pits”, probably caused by parasitism, is found in nine species of five genera, mostly in tornoceratids. The Sand fauna indicates that the species origination rate in tornoceratids remained high after the Lower Kellwasser Event.

Planarity and non-separating cycles in uniform high genus quadrangulations

We study large uniform random bipartite quadrangulations whose genus grows linearly with the number of faces. Their local convergence was recently established by Budzinski and the author [9, 10]. Here we study several properties of these objects which are not captured by the local topology. Namely we show that balls around the root are planar with high probability up to logarithmic radius, and we prove that there exist non-contractible cycles of constant length with positive probability.

Free boundary minimal surfaces in the unit three-ball via desingularization of the critical catenoid and the equatorial disc

We construct a new family of high genus examples of free boundary minimal surfaces in the Euclidean unit 3-ball by desingularizing the intersection of a coaxial pair of a critical catenoid and an equatorial disc. The surfaces are constructed by singular perturbation methods and have three boundary components. They are the free boundary analogue of the Costa–Hoffman–Meeks surfaces and the surfaces constructed by Kapouleas by desingularizing coaxial catenoids and planes. It is plausible that the minimal surfaces we constructed here are the same as the ones obtained recently by Ketover by using the min-max method.

High-genus multicompartment vesicles evolved from large compound micelles

We report a high-genus multicompartment vesicle (HGMV) that is self-assembled from poly(ethylene oxide)-block-poly[4-azophenyl-(2-carbamoyloxymethyl)ethyl methacrylate] (PEO43-b-PACEMA10). The formation process of these HGMVs was investigated by transmission electron microscopy (TEM), fluorescence spectroscopy,...