Jones polynomial invariants for knots and satellites

H. R. Morton, P. Strickland
1991 Mathematical proceedings of the Cambridge Philosophical Society (Print)  
Results of Kirillov and Reshetikhin on constructing invariants of framed links from the quantum group SU(2) g 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) g -invariant of a link L is a linear combination of Jones polynomials of parallels of L, where the combination is determined explicitly from
more » ... ned 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. H. R. MORTON AND P. STRICKLAND In particular, we can readily deduce the result due to Yamada[17]^ on the Jones polynomial of 2-parallels. Having done this, we shall present the results on satellite knots in a form which is strikingly reminiscent of the classical Fox formula for Alexander polynomials, and which gives a key role to the Hopf link and its invariants. We give explicit finiteness conditions limiting the number of invariants of the companion and pattern which are needed in calculating a given invariant for the satellite. Review of work of Kirillov and Beshetikhin In the papers [12] and [13], the foundations of methods to calculate regular isotopy invariants of link diagrams corresponding to finite-dimensional representations of quantum groups are set out. The idea is to set up a functor relating tangles to module homomorphisms, following a scheme of Turaev as set out in a talk at the Sussex meeting in 1987; see [14] and [5] . In this scheme oriented tangles correspond to homomorphisms of finite-dimensional modules over a given quantum group, in such a way that the natural tangle operations of composition and 'horizontal' adjoining correspond to composition and tensor products of modules. (In order to give meaning to the tensor products, it is essential to use the coproduct for the Hopf algebra structure of the quantum group.) Additional relations, including the Yang-Baxter equation, allow us to use this correspondence to define link invariants, by considering either (0,0)-or (1, l)-tangles; this will be made clear later. The paper [7] takes the programme outlined above, and fills in many details for the case of the quantum group SU(2) 9 . For any quantum group arising from a simple Lie algebra, it has been proved, [15] , that the representations (which will always be taken to be finite-dimensional from now on) are in 1-1 correspondence with those of the original Lie algebra. In the case of su(2), there is one representation in every dimension, indexed in [7] by half-integer 'spins'. Turaev's general construction may be described for SU(2) g as follows. For each tangle diagram T we assign a finite-dimensional irreducible representation of SU(2) g to each string, or in other words an irreducible module over SU(2) g . We associate an SU(2) 9 module to the top of the tangle T by taking the tensor product of the modules assigned to the strings, in order, which meet the top of T. We can similarly associate a tensor product of modules with the bottom of T using the modules of the strings which meet the bottom. The goal is then to produce a module homomorphism, determined by T and the choice of modules on the strings, from the top module to the bottom module, as indicated in Figure 1 . These homomorphisms will be chosen functorially so that when two tangles S and T are composed to form the tangle ST, and the assignment of modules to strings is compatible, then the module homomorphisms for S and T compose to give the homomorphism for ST. We also arrange that when tangles are adjoined side by side the homomorphism for the new tangle is the tensor product of the homomorphisms for its constituents. In this general framework the module associated to the top of a tangle where no strings are incident is the trivial module, which may be equated with the ground ring or field A. In what follows, we take A to contain at least the Laurent polynomial ring /[g^*], where the deformation parameter q in the quantum group is
doi:10.1017/s0305004100069589 fatcat:tysnn7spwrcphltgnm7cp2ebgu