Before revision:  This is the first nontrivial construction of 2-hop spanners. (IV) For every sufficiently large  n, there exists a set P of n points on a circle, such that every plane hop spanner on P has hop stretch factor at least 4. Previously, no lower bound greater than 2 was known. (V) For every  point set on a circle, there exists a plane  4-hop spanner.
After revision:  This is the first nontrivial construction of 2-hop spanners. (IV) For every sufficiently large positive integer n, there exists a set P of n points on a circle, such that every plane hop spanner on P has hop stretch factor at least 4. Previously, no lower bound greater than 2 was known. (V) For every point set on a circle, there exists a plane 4-hop spanner.