Combinatorics of embeddings

We offer the following explanation of the statement of the Kuratowski graph planarity criterion and of 6/7 of the statement of the Robertson-Seymour-Thomas intrinsic linking criterion. Let us call a cell complex 'dichotomial' if to every cell there corresponds a unique cell with the complementary set of vertices. Then every dichotomial cell complex is PL homeomorphic to a sphere; there exist precisely two 3-dimensional dichotomial cell complexes, and their 1-skeleta are K_5 and K_{3,3}; and precisely six 4-dimensional ones, and their 1-skeleta all but one graphs of the Petersen family. In higher dimensions n>2, we observe that in order to characterize those compact n-polyhedra that embed in S^{2n} in terms of finitely many "prohibited minors", it suffices to establish finiteness of the list of all (n-1)-connected n-dimensional finite cell complexes that do not embed in S^{2n} yet all their proper subcomplexes and proper cell-like combinatorial quotients embed there. Our main result is that this list contains the n-skeleta of (2n+1)-dimensional dichotomial cell complexes. The 2-skeleta of 5-dimensional dichotomial cell complexes include (apart from the three joins of the i-skeleta of (2i+2)-simplices) at least ten non-simplicial complexes.Comment: 49 pages, 1 figure. Minor improvements in v2 (subsection 4.C on transforms of dichotomial spheres reworked to include more details; subsection 2.D "Algorithmic issues" added, etc

Semi-aural Interfaces: Investigating Voice-controlled Aural Flows

To support mobile, eyes-free web browsing, users can listen to ‘playlists’ of web content— aural flows . Interacting with aural flows, however, requires users to select interface buttons, tethering visual attention to the mobile device even when it is unsafe (e.g. while walking). This research extends the interaction with aural flows through simulated voice commands as a way to reduce visual interaction. This paper presents the findings of a study with 20 participants who browsed aural flows either through a visual interface only or by augmenting it with voice commands. Results suggest that using voice commands reduced the time spent looking at the device by half but yielded similar system usability and cognitive effort ratings as using buttons. Overall, the low-cognitive effort engendered by aural flows, regardless of the interaction modality, allowed participants to do more non-instructed (e.g. looking at the surrounding environment) than instructed activities (e.g. focusing on the user interface)