Integrality of Homfly (1, 1)-tangle invariants

Given an invariant J(K) of a knot K, the corresponding (1, 1)-tangle invariant J ′(K) = J(K)/J(U) is defined as the quotient of J(K) by its value J(U) on the unknot U . We prove here that J ′ is always an integer 2-variable Laurent polynomial when J is the Homfly satellite invariant determined by decorating K with any eigenvector of the meridian map in the Homfly skein of the annulus. Specialisation of the 2-variable polynomials for suitable choices of eigenvector shows that the (1, 1)-tangle irreducible quantum sl(N) invariants of K are integer 1-variable Laurent polynomials.