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The main goal is to find the Homfly polynomial of a link formed by decorating each component of the Hopf link with the closure of a directly oriented tangle. Such decorations are spanned in the Homfly skein of the annulus by elements Q_λ, depending on partitions λ. We show how the 2-variable Homfly invariant <λ, μ> of the Hopf link arising from decorations Q_λ and Q_μ can be found from the Schur symmetric function s_μ of an explicit power series depending on λ. We show also that the quantum invariant of the Hopf link coloured by irreducible sl(N)_q modules V_λ and V_μ, which is a 1-variable specialisation of <λ, μ>, can be expressed in terms of an N × N minor of the Vandermonde matrix (q^ij).

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.

We give new examples of 2-component links with linking number one and unknotted components that are topologically concordant to the positive Hopf link, but not smoothly so – in addition, they are not smoothly concordant to the positive Hopf link with a knot tied in the first component. Such examples were previously constructed by Cha–Kim–Ruberman–Strle; we show that our examples are distinct in smooth concordance from theirs.

In this note we define the Hopf-braid group, a group that is directly related to the group of motions of n mutually distinct lines through the origin in , which is better known as the braid group of the two-sphere. It is also related to the motion group of the Hopf link in the three-sphere. This relationship is provided by considering the link of a union of complex lines through the origin in (i.e. the intersection of the lines with the unit 3-sphere centered at the origin in ). Through the study of this group we also illustrate some of the connections between the field of knots and braids and that of hyperplane arrangements.