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On the invariants of torus knots derived from quantum groups. (English) Zbl 0787.57006
This paper gives a general formula for quantum invariants of torus knots. The formula covers invariants for an irreducible representation of any quantum group associated to a classical Lie algebra. It involves a mix of the representation theory for the classical algebra and for $$S_ n$$, and in particular a cyclic subgroup of order $$n$$, when dealing with an $$(m,n)$$ torus knot.
As an example the authors use the fundamental representation of $$SU(N)_ q$$ as $$N$$ varies, and compare the results with previous calculations of V. F. R. Jones for the Homfly polynomial of torus knots in [Ann. Math., II. Ser. 126, 335-388 (1987; Zbl 0631.57005)].
A slight reformulation of the results could lead to corresponding formulae for invariants of $$(m,n)$$ cables about a knot $$C$$ in terms of the invariants of $$C$$, as suggested in an unpublished paper of P. M. Strickland [On the quantum enveloping algebra invariants of cables, Preprint, Liverpool University, 1990].

##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010) 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory 17B37 Quantum groups (quantized enveloping algebras) and related deformations
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