Let $M^{ reg}$ be the topological moduli space of long knots up to regular isotopy, and for any natural number $n > 1$ let $M^{ reg}_n$ be the moduli space of all n-cables of framed long knots which are twisted by a string link to a knot in the solid torus $V^3$ . We upgrade the Vassiliev invariant $v_2$ of a knot to an integer valued combinatorial 1-cocycle for $M^{ reg}_n$ by a very simple formula. This 1-cocycle depends on a natural number $a \in \mathbb{ Z}\cong H_1(V^3;\mathbb{ Z})$ with $0<a<n$ as a parameter and we obtain a polynomial-valued 1-cocycle by taking the Lagrange interpolation polynomial with respect to the parameter. We show that it induces a non-trivial pairing on $H_0(M^{ reg}_n) \times H_0(M^{ reg})$ already for $n=2$.
Categories: math.GT