zbMATH — the first resource for mathematics

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formulae for invariants of \((m,n)\) cables about a knot \(C\) in terms of the invariants of \(C\), as suggested in an unpublished paper of P. M. Strickland [On the quantum enveloping algebra invariants of cables, Preprint, Liverpool University, 1990].

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
Full Text: DOI