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Invariants of 3-manifolds derived from universal invariants of framed links. (English) Zbl 0859.57018
A quantum invariant of a link is defined if we are given a ribbon Hopf algebra and its representations. A universal link invariant, introduced first by R. J. Lawrence [A universal link invariant using quantum groups, in ‘Differential geometric methods in theoretical physics’ (Chester, 1998), 211-232] and modified by the author [J. Knot Theory Ramifications 2, 211-232 (1993; Zbl 0798.57006)], is an invariant defined by using only a ribbon Hopf algebra (without using its representations).
In this paper under review, the author gives a method to construct invariants directly from a ribbon Hopf algebra without using representations. It depends on a map \(\chi: A \to \mathbb{C}\), where \(A\) is a ribbon Hopf algebra and \(\chi\) satisfies the condition \(\chi (ab) = \chi (ba)\). This construction was first pointed out by M. A. Hennings [Invariants of links and 3-manifolds obtained from Hopf algebras (preprint)]. The case where \(A\) is finite is also studied by L. H. Kauffman and D. E. Radford [Invariants of 3-manifolds derived from finite dimensional Hopf algebras (preprint)].
After a general definition, the case where \(A=U_q (\text{sl}_2)\) is studied in detail. Unfortunately it is not known whether these universal invariants are independent of quantum invariants though it is shown that they cannot be a linear combination of invariants of framed links.
Reviewer: H.Murakami (Tokyo)

MSC:
57N10 Topology of general \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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References:
[1] DOI: 10.1007/BF01239527 · Zbl 0725.57007 · doi:10.1007/BF01239527
[2] DOI: 10.1142/S0218216593000131 · Zbl 0798.57006 · doi:10.1142/S0218216593000131
[3] Lawrence, Differential geometric methods in theoretical physics pp 55– (1988)
[4] Drinfeld, Algebra and Analysis 1 pp 30– (1989)
[5] DOI: 10.1007/BF01406222 · Zbl 0377.55001 · doi:10.1007/BF01406222
[6] DOI: 10.1016/0040-9383(79)90010-7 · Zbl 0413.57006 · doi:10.1016/0040-9383(79)90010-7
[7] DOI: 10.1007/BF01232277 · Zbl 0745.57006 · doi:10.1007/BF01232277
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