A basis for the full Homfly skein of the annulus

Abstract

The meridian maps of the full Homfly skein of the annulus are linear endomorphisms induced by the insertion of a meridian loop, with either orientation, around a diagram in the annulus. The eigenvalues of the meridian maps are known to be distinct, and are indexed by pairs of partitions of integers p and n into k and k* parts respectively. We give here an explicit formula for a corresponding eigenvector as the determinant of a (k*+k)x(k*+k) matrix whose entries are skein elements corresponding to partitions with a single part. This extends the results of Kawagoe and Lukac, for the case p=0, giving a basis for the subspace of the skein spanned by closed braids all oriented in the same direction. Their formula uses the Jacobi-Trudy determinants for Schur functions in terms of complete symmetric functions. Our matrices have a similar pattern of entries in k rows, and a modified pattern in k* rows, resulting in a combination of closed braids with up to n strings oriented in one direction and p in the reverse direction. We discuss the 2-variable knot invariants resulting from decoration of a knot by these skein elements, and their relation to unitary quantum invariants.