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 ).

The Hecke algebra H n contains well known idempotents E λ which are indexed by Young diagrams with n cells. They were originally described by Gyoja [5].A skein theoretical description of E λ was given by Aiston and Morton [2]. The closure of E λ becomes an element Q λ of the skein of the annulus. In this skein, they are known to obey the same multiplication rule as the symmetric Schur functions s λ as stated in theorem 8.2. But previous proofs of this fact as in [1] used results about quantum groups which were far beyond the scope of skein theory.Our elementary proof was motivated by [6] and uses only skein theory and basic algebra. The skein of a planar surfaceWe consider a planar surface F with designated n incoming and n outgoing boundary points for some integer n ≥ 0. The Homfly skein S(F ) is defined as the module of linear combinations of oriented tangles in F quotiented by regular isotopy (i.e. Reidemeister moves II and III), the two local skein relations in figure 1 where v and s are variables, and the relation that a disjoint simple closed curve can be removed from a diagram at the expense of multiplication with the scalar δ =