Cyclic order and dissection order of certain arcs

Abstract:

Let arc $A$ in the conformal plane or on the sphere have local cyclic order three and cyclic order $t$. It can be decomposed into a finite number of subarcs of cyclic order three. Let the disection order of $A$ be the minimum number of arcs in such a decomposition. The principal result of this paper is that the cyclic order $t$ and dissection order $d$ of $A$ satisfy the relation $d + 2 \leqq t \leqq 3d$. In establishing this result it is proved that a necessary and sufficient condition, that an arc of local cyclic order three shall be of global cyclic order three, is that there exists a circle meeting it only at the endpoints.