Satellites and surgery invariants. (English) Zbl 0763.57006

Knots 90, Proc. Int. Conf. Knot Theory Rel. Topics, Osaka/Japan 1990, 47-66 (1992).
[For the entire collection see Zbl 0747.00039.]
The \(SU(2)_ q\) invariants of a framed \(k\)-component link \(L\), when the variable \(q\) is set equal to an \(r\)th root of unity, yield a map \(J_ r(L):R_ r^{\otimes k}\to\Lambda_ r\), where \(R_ r\) is a finite- dimensional truncation of the representation ring of \(SU(2)\) and \(\Lambda_ r=\mathbb{Z}[e^{\pi i/2r}]\). A formula relating the quantum invariants of a satellite knot with those of its companion and pattern links is used to exhibit Reshetikhin and Turaev’s invariant of the 3- manifold given by surgery on \(L\), as the evaluation of \(J_ r(L)\) on a fixed element in each \(R_ r\). It is shown how the invariant of the manifold given by Dehn surgery with coefficients \(a_ i/b_ i\) on a link \(L\) can be found by evaluating \(J_ r(L)\) on certain elements \(M_{a_ i/b_ i}\in R_ r\).
Reviewer: C.Kearton (Durham)


57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory


Zbl 0747.00039