On surgery along Brunnian links in $3$–manifolds

Abstract

We consider surgery moves along $\left(n+1\right)$–component Brunnian links in compact connected oriented $3$–manifolds, where the framing of the components is in $\left\{\frac{1}{k}\mathrm{\text{; }}k\in Z\right\}$. We show that no finite type invariant of degree $<2n-2$ can detect such a surgery move. The case of two link-homotopic Brunnian links is also considered. We relate finite type invariants of integral homology spheres obtained by such operations to Goussarov–Vassiliev invariants of Brunnian links.