THE ZONE MODULUS OF A LINK

In this paper, we construct a conformally invariant functional for two-component links called the zone modulus of the link. Its main property is to give a sufficient condition for a link to be split.

The zone modulus is a positive number, and its lower bound is 1. To construct a link with modulus arbitrarily close to 1, it is sufficient to consider two small disjoint spheres each one far from the other and then to construct a link by taking a circle enclosed in each sphere. Such a link is a split link. The situation is different when the link is non-split: we will prove that the modulus of a non-split link is greater than . This value of the modulus is realized by a special configuration of linked circles called the Clifford link.

As a corollary, we show that if the thickness of a non-split two-component link embedded in S³ is equal to , then the link is the standard geometric Hopf link.