A low-dimensional version of our main result is the following `converse' of the Conway-Gordon-Sachs Theorem on intrinsic linking of the graph $K_6$ in 3-space: For any integer $z$ there are 6 points $1,2,3,4,5,6$ in 3-space, of which every two $i,j$ are joint by a polygonal line $ij$, the interior of one polygonal line is disjoint with any other polygonal line, the linking coefficient of any pair disjoint 3-cycles except for $\{123,456\}$ is zero, and for the exceptional pair $\{123,456\}$ is $2z+1$. We prove a higher-dimensional analogue, which is a `converse' of a lemma by Segal-Spie\.z.

中文翻译：

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与内在联系定理的一些“相反”

我们的主要结果的低维版本是 Conway-Gordon-Sachs 定理在 3 空间中图 $K_6$ 的内在链接的以下“逆”：对于任何整数 $z$ 有 6 个点 $1,2 ,3,4,5,6$在3空间中，其中每两个$i,j$由一条折线$ij$连接，一条折线的内部与任何其他折线不相交，连接系数除了 $\{123,456\}$ 之外，任何不相交的 3-cycles 的对都为零，对于特殊对 $\{123,456\}$ 是 $2z+1$。我们证明了一个更高维的类似物，它是 Segal-Spie\.z 引理的“逆”。