Vassiliev invariants and the Poincaré conjecture (Q5917539)

\textit{X.-S. Lin} [Topology 33, No. 1, 45--71 (1994; Zbl 0816.57013)] proved that the module of finite type invariants of links in a \(3\)-manifold \(M\) with \(\pi_1(M)=\pi_2(M)=0\) (i.e., a contractible \(3\)-manifold or a homotopy sphere) is isomorphic to that for the \(3\)-sphere, up to possible \(2\)-torsion. Lin's result does not provide an example of a pair of two knots in \(M\) which are not distinguishable by Vassiliev invariants. The paper under review shows that there is a pair of knots in \(M\) indistinguishable by any Vassiliev invariants, if \(M\) is any Whitehead manifold, i.e., an open, contractible \(3\)-manifold which is not homeomorphic to \(\mathbb{R}^3\) but is embeddable into \(\mathbb{R}^3\). It is also proved that if Vassiliev invariants distinguish all knots in each homotopy sphere, then the Poincaré conjecture is true. To prove these results, the author uses the following fact: if \(M\) is a simply connected \(3\)-manifold and \(h\) is an orientation-preserving self-embedding of \(M\), then for any knot \(K\) in \(M\), \(K\) and its image \(h(K)\) are not distinguishable by any Vassiliev invariants of knots in \(M\).