#### Volume 7, issue 1 (2007)

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Integrality of Homfly 1–tangle invariants

### H R Morton

Algebraic & Geometric Topology 7 (2007) 327–338
 arXiv: math.GT/0606336
##### 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.

##### Keywords
Homfly, skein, annulus, quantum $sl(N)$, irreducible, integrality, 1–tangle
##### Mathematical Subject Classification 2000
Primary: 57M25, 57M27
Secondary: 57R56