Integrality of Homfly (1,1)-tangle invariants

Details Integrality of Homfly (1,1)-tangle invariants Integrality of Homfly (1,1)-tangle invariants

2006-06-14

arxiv.org/abs/math/0606336v1

Algebraic and Geometric Topology, 7 (2007), 227-238.

Mathematics

Geometric Topology

10 pages, including several interspersed figures

Scientific paper

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.

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