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


Given an invariant J(K) of a knot K, the corresponding 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 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 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(N) invariants of K are integer 1–variable Laurent polynomials.

Homfly, skein, annulus, quantum $sl(N)$, irreducible, integrality, 1–tangle
Mathematical Subject Classification 2000
Primary: 57M25, 57M27
Secondary: 57R56
Received: 18 December 2006
Accepted: 12 February 2007
Published: 29 March 2007
H R Morton
Department of Mathematical Sciences
University of Liverpool
Peach Street
Liverpool L69 7ZL