Polynomial invariants are polynomial (Q1902217)

For a fixed Vassiliev knot invariant \(V\) of type \(m\) with values in a normed vector space, and a singular knot \(K\) with \(k\) double points and a large number \(n\) of crossings, it is shown that \(V(K)\) is bounded by \(C_k n^{m-k}\) for some fixed constants \(C_k\). This includes the case \(k = 0\) of non-singular knots and can be interpreted as saying that polynomial invariants grow polynomially. Also, \(V(K)\) can be computed (in terms of \(V\) of finitely many fixed knots) in approximately \(n^m\) computational steps. As stated in the introduction, a bound by a polynomial of degree \(2m\) resp. computability in high-degree polynomial time is easier to show. For the proof by downward induction on \(k\), it is shown that two \(k\)-singular knot projections as above having the same underlying degree \(k\) chord diagram can be connected by a nicely controlled path of singular knots containing only finitely many \((k+1)\)-singular knots.