Quantum invariants of 3-manifolds (Q2776342)

The paper is an introduction to quantum invariants of 3-manifolds using linear skein theory. Sections 2, 3 and 4 are devoted to proving the existence of the quantum \(SU_q(2)\) invariants of closed oriented \(3\)-manifolds using the Kirby calculus on framed links and the Kauffman bracket linear skein theory. This approach to the \(SU_q(2)\) invariants has been given by the author in several papers [Pac. J. Math. 149, No. 2, 337-347 (1991; Zbl 0728.57011); Math. Ann. 290, No. 4, 657-670 (1991; Zbl 0739.57004); Comment. Math. Helv. 67, No. 4, 571-591 (1992; Zbl 0779.57008); J. Knot Theory Ramifications 2, No. 2, 171-194 (1993; Zbl 0793.57003)]. Section 5 deals with relationships between the \(SU_q(2)\) invariants and some geometric invariants, such as the genus of a \(3\)-manifold and the generalized bridge number and the tunnel number of a knot in a \(3\)-manifold. In Section 6 the \(SU_q(2)\) invariant of the product \(S^1\times F\) of a circle \(S^1\) and a closed orientable surface \(F\) is calculated using the 6j-symbols and the Clebsch-Gordan formula. Section 7 describes \textit{J. Roberts}'s work [Topology 34, No. 4, 771-787 (1995; Zbl 0866.57014)] that relates the \(SU_q(2)\) invariants and the Turaev-Viro invariants [\textit{V. G. Turaev} and \textit{O. Y. Viro}, ibid. 31, No. 4, 865-902 (1992; Zbl 0779.57009)] of \(3\)-manifolds. Section 8 describes the quantum \(SU_q(N)\) invariants of \(3\)-manifolds by \textit{V. Turaev} and \textit{H. Wenzl} [Int. J. Math. 4, No. 2, 323-358 (1993; Zbl 0784.57007)] in \textit{Y. Yokota}'s approach [Math. Ann. 307, No.1, 109-138 (1997; Zbl 0953.57009)] based on the HOMFLY skein theory.NEWLINENEWLINEFor the entire collection see [Zbl 0977.00029].