Finite type link invariants of 3-manifolds
Given an invariant of oriented links in an oriented 3-manifold one can extend it to an invariant of oriented singular links with one double point by taking the difference of the invariant of the two nearby non- singular oriented links with an appropriate sign. Inductively one obtains an invariant of oriented singular links with finitely many double points. The author gives conditions when an invariant of singular links with \(n+1\) double points is obtained in this way from an invariant of singular links with \(n\) double points. This involves studying 2-parameter families of maps from 1-dimensional polyhedra into 3-manifolds \(M\). In the case when \(\pi_ 1 (M)= \pi_ 2 (M)=0\) or when \(\pi_ 2 (M)=0\), \(\pi_ 1(M)\) is finite, and the ring where the invariant takes its values is torsion free he can give ``local'' necessary and sufficient conditions. He uses this result to give an intrinsic definition of the Jones polynomial and to prove the following result. Suppose that \(M\subset M'\) are oriented 3-manifolds with \(\pi_ 1= \pi_ 2 =0\) and let \(V_ m (M)\) be the \(R\)-module of \(R\)-valued Vassiliev-invariants of links in \(M\) of order \(m\) (i.e. invariants whose extensions vanish on singular links with at least \(m+1\) double points). Assume further that \(R\) has no 2- torsion. Then the inclusion \(M \hookrightarrow M'\) induces an isomorphism \(V_ m(M)\to V_ m (M')\).
oriented links in an oriented 3-manifold
maps from 1-dimensional polyhedra into 3-manifolds
scientific article; zbMATH DE number 527437