Evaluations of the 3-manifolds invariants of Witten and Reshetikhin- Turaev for \(sl(2,C)\). (English) Zbl 0756.57006

Kirby, Robion; Melvin, Paul

Geometry of low-dimensional manifolds. 2: Symplectic manifolds and Jones- Witten-Theory, Proc. Symp., Durham/UK 1989, Lond. Math. Soc. Lect. Note Ser. 151, 101-114 (1990).

[For the entire collection see Zbl 0722.00024.]
Fix an integer \(r>1\). The 3-manifold invariant \(\tau_ r\) assigns a complex number \(\tau_ r(M)\) to each oriented, closed, connected 3- manifold \(M\) and satisfies: (1) \(\tau_ r(M\#N)=\tau_ r(M)\cdot\tau_ r(N)\); (2) \(\tau_ r(-M)=\overline{\tau_ r(M)}\); (3) \(\tau_ r(S^ 3)=1\).
\(\tau_ r(M)\) is defined as a weighted average of colored, framed link invariants \(J_{L,k}\) of a framed link \(L\) for \(M\), where a coloring of \(L\) is an assignment of integers \(k_ i\), \(0<k_ i<r\), to the components \(L_ i\) of \(L\). The \(k_ i\) denote representations of the Hopf algebra of dimension \(k_ i\), and \(J_{L,k}\) is a generalization of the Jones polynomial of \(L\) at \(q\). The authors describe the Reshetikhin-Turaev version of \(\tau_ r\) for \(q=e^{2\pi i/r}\), giving a cabling formula, a symmetry principle, and evaluations at \(r=3,4\) and \(6\).

Reviewer: A.K.Guts (Omsk)

MSC:

57N10	Topology of general \(3\)-manifolds (MSC2010)
57M99	General low-dimensional topology
82B23	Exactly solvable models; Bethe ansatz
81R50	Quantum groups and related algebraic methods applied to problems in quantum theory

Keywords:

3-manifold invariant of Witten; quantum group invariants of 3-manifolds; Kirby calculus; framed link; cabling formula; symmetry principle

Citations:

Zbl 0722.00024