Integrality of Homfly 1–tangle invariants

Abstract

Given an invariant $J\left(K\right)$ of a knot $K$, the corresponding 1–tangle invariant $J^{\prime }\left(K\right)=J\left(K\right)∕J\left(U\right)$ is defined as the quotient of $J\left(K\right)$ by its value $J\left(U\right)$ on the unknot $U$. We prove here that 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 then $J^{\prime }$ is always an integer 2–variable Laurent polynomial. Specialisation of the 2–variable polynomials for suitable choices of eigenvector shows that the 1–tangle irreducible quantum $sl\left(N\right)$ invariants of $K$ are integer 1–variable Laurent polynomials.