Mathematics – Geometric Topology
Scientific paper
2004-08-05
Math. Proc. Camb. Philos. Soc. 141 (2006), 81-100
Mathematics
Geometric Topology
27 pages, with about 20 embedded figures. Two misprints in the polynomials for the trefoil and figure eight knots in the last
Scientific paper
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.
Hadji Richard J.
Morton Hugh R.
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