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Colored ribbon Hopf algebras and universal invariants of framed links. (English) Zbl 0798.57006
The author introduces a notion of a colored ribbon Hopf algebra, which is a deformation of a ribbon Hopf algebra, and defines a universal invariant of a framed link. A universal invariant was first introduced by R. J. Lawrence [A universal link invariant using quantum groups, in ‘Differential geometric methods in theoretical physics’, Chester, 473-545 (1988)] but the invariant here is a little different since the quantum trace is taken into account. He gives an example of a colored ribbon Hopf algebra which is a quotient of \(U_ q(\text{sl}_ 2)\) giving a universal \(R\)-matrix explicitly. The universal invariant constructed from this generalizes both the framed link invariants given in [N. Yu. Reshetikhin and V. G. Turaev, Commun. Math. Phys. 127, No. 1, 1-26 (1990; Zbl 0768.57003)] and those given in [Y. Akutsu, T. Deguchi and T. Ohtsuki, J. Knot Theory Ramifications 1, No. 2, 161-184 (1992; Zbl 0758.57004)]. Note that the former includes the Jones polynomial and the latter the multi-variable Alexander polynomial.
Unfortunately, there are no examples which confirm that his invariant is essentially new.
Reviewer: H.Murakami (Osaka)

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
82B23 Exactly solvable models; Bethe ansatz
17B37 Quantum groups (quantized enveloping algebras) and related deformations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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