Let Mreg be the topological moduli space of long knots up to regular isotopy, and for any natural number n > 1 let Mnreg 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 V3. We upgrade the Vassiliev invariant v2 of a knot to an integer valued combinatorial 1-cocycle for Mnreg by a very simple formula. This 1-cocycle depends on a natural number a ∈ ℤ≅H1(V3; 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 H0(Mnreg) × H 0(Mreg) already for n = 2.

中文翻译：

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更多经典结的 1-cocycles

让米注册是直到正则同位素的长结的拓扑模空间，并且对于任何自然数n > 1让米n注册是所有的模空间n- 框架长结的缆绳，由绳索扭绞，连接到实心圆环中的结五3. 我们升级 Vasiliev 不变量v2一个结到一个整数值的组合 1-cocycle 为米n注册通过一个非常简单的公式。这个 1-cocycle 取决于一个自然数一种 ∈ ℤ≅H1(五3; ℤ)和0 < 一种 < n作为参数，我们通过对参数进行拉格朗日插值多项式来获得多项式值的 1-cocycle。我们证明它在H0(米n注册) × H 0(米注册)已经为n = 2.