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Let   m   be the closure of the Hecke algebra with  m  strings,  H   m  , in the oriented framed Homfly skein  of the annulus, which provides the natural parameter space for the Homfly satellite invariants of a knot. The submodule  +  ⊂  spanned by the union ⋃  m  ⩾ 0    m   is an algebra, isomorphic to the algebra of the symmetric functions. Turaev's geometrical basis for  +  consists of monomials in closed  m ‐braids  A   m  , the closure of the braid σ  m −1 ·…·σ 2 σ 1 . We collect and expand formulae relating elements expressed in terms of symmetric functions to Turaev's basis. We reformulate the formulae of Rosso and Jones for quantum sl( N ) invariants of cables in terms of plethysms of symmetric functions, and use the connection between quantum sl( N ) invariants and the skein  +  to give a formula for the satellite of a cable as an element of the Homfly skein  + . We can then analyse the case where a cable is decorated by the pattern  P   d   which corresponds to a power sum in the symmetric function interpretation of  +  to get direct relations between the Homfly invariants of some diagrams decorated by power sums

Geometrical relations and plethysms in the Homfly skein of the annulus