Embeddability of arrangements of pseudocircles into the sphere

Abstract

An arrangement of pseudocircles is a finite set of oriented closed Jordan curves each two of which cross each other in exactly two points. To describe the combinatorial structure of arrangements on closed orientable surfaces, in [J. Linhart, R. Ortner, On the combinatorial structure of arrangements of oriented pseudocircles, Electron. J. Combin. 11 (2004). Research Paper 30, 13 pages (electronic)] so-called intersection schemes were introduced. Building upon results about the latter, we first clarify the notion of embedding of an arrangement. Once this is done, it is shown how the embeddability of an arrangement depends on the embeddability of its subarrangements. The main result presented is that an arrangement of pseudocircles can be embedded into the sphere if and only if all of its subarrangements of four pseudocircles are embeddable into the sphere as well.