Integrality of Homfly (1,1)-tangle invariants

Abstract

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.

10 pages, including several interspersed figures

References in corpus (3)

• Idempotents of Hecke algebras of type A
• A family of embedding spaces
• Homfly Polynomials of Generalized Hopf Links

Cited by in corpus (7)

• The HOMFLYPT skein algebra of the torus and the elliptic Hall algebra
• String theory and the Kauffman polynomial
• Checks of integrality properties in topological strings
• Composite Representation Invariants and Unoriented Topological String Amplitudes
• Composite Invariants and Unoriented Topological String Amplitudes
• Relations between Kauffman and Homfly satellite invariants
• A simple proof of the strong integrality for full colored HOMFLYPT invariants