  The image of a polygonal knot K under a spherical inversion of R^3 (union
infinity) is a simple closed curve made of arcs of circles, having the same
knot type as the mirror image of K. Suppose we reconnect the vertices of the
inverted polygon with straight lines, making a new polygon. This may be a
different knot type. For example, a certain 7-segment figure-8 knot can be
transformed to a figure-8 knot, right and left handed trefoils, or an unknot,
by selecting different inverting spheres. Which knot types can be obtained from
a given original polygon K under this process? We show that for large n, almost
all n-segment knot types cannot be reached from one initial n-segment polygon,
using a single inversion or even the whole Mobius group.
  The number of knot types arising from an n-vertex polygon is bounded by the
number of complementary domains of a certain system of n(n-3)/2 round 2-spheres
in R^3. We show the number of domains is at most polynomial in the number of
spheres. In the analysis, we obtain an exact formula for the number of
complementary domains. On the other hand, the number of knot types that can be
represented by n-segment polygons is exponential in n.
