论文标题
更多的1个古典结
More 1-cocycles for classical knots
论文作者
论文摘要
令$ m^{reg} $成为长期结的拓扑模量空间，直到常规同位素，对于任何天然数字$ n> 1 $ ling $ m^{reg} _n $是所有框架长结的模量空间，这些空间是由固体torus $ v^3 $中的弦乐链接扭曲的链条链路扭曲的。我们通过一个非常简单的公式升级了一个打结的Vassiliev $ v_2 $，以$ m^{reg} _n $的$ m^{reg} _n $。这1个循环依赖于\ Mathbb {z} \ cong H_1（v^3; \ Mathbb {z}）$的天然数字$ a \ in $ 0 <a <n $作为参数，我们获得了与lagrange interpolation interpolation toplenomial相关的多项式价值1 cocycle。我们表明，它在$ h_0（m^{reg} _n）\ times h_0（m^{reg}）$上诱导了一个非平凡的配对，以$ n = 2 $。
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 $n=2$.
