## 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)

### MSC:

 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