More 1-cocycles for classical knots

Let $M^{reg}$ be the topological moduli space of long knots up to regular isotopy, and for any natural number $n>1$ let $M_n^{reg}$ 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 $M_n^{reg}$ 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 $H_0(M_n^{reg})\times H_0(M^{reg})$ already for $n=2$.