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Vassiliev invariants and the Poincaré conjecture. (English) Zbl 1052.57016

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 \(M\).

MSC:

57M40 Characterizations of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
57N35 Embeddings and immersions in topological manifolds

Citations:

Zbl 0816.57013
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