Resumen de Geometrical relations and plethysms in the Homfly skein of the annulus

H. R. Morton, Pedro María González Manchón

• Let m be the closure of the Hecke algebra with m strings, Hm, 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 Am, the closure of the braid m�1·...·21. 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 Pd 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