Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Jones polynomial invariants for knots and satellites

H. R. Mortona1 and P. Stricklanda1

a1 Department of Pure Mathematics, University of Liverpool, PO Box 147, Liverpool L69 3BX

Abstract

Results of Kirillov and Reshetikhin on constructing invariants of framed links from the quantum group SU(2)q are adapted to give a simple formula relating the invariants for a satellite link to those of the companion and pattern links used in its construction. The special case of parallel links is treated first. It is shown as a consequence that any SU(2)q-invariant of a link L is a linear combination of Jones polynomials of parallels of L, where the combination is determined explicitly from the representation ring of SU(2). As a simple illustration Yamada's relation between the Jones polynomial of the 2-parallel of L and an evaluation of Kauffman's polynomial for sublinks of L is deduced.

(Received January 02 1990)

(Revised May 04 1990)