Here is a picture of 6 topological circles which intersect pairwise in two points,
but which cannot be isotoped to be round circles. That is, the circles can't
be moved to be round, without changing the pattern of intersections. If
you delete any circle, though, the five left can be made round.
 

Andrew Cassson showed that if you make each of the circles bound
disks in upper half-space, then there will always be a
``football region'' (American football), which consists of
a region which looks like a football, with three planes adjacent to
it. This is significant for understanding the patterns of
incompressible surfaces in the universal covers of 3-dimensional
manifolds.

Reference: Configurations of curves and geodesics on surfaces,
Joel Hass and Peter Scott, preprint