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 joined by a polygonal line $ij$, the interior of one polygonal line is disjoint with any other polygonal line, the linking coefficient of any pair of 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.
翻译:我们主要结果的一个低维版本是Conway-Gordon-Sachs理论的以下“反方”:图形在3个空格中的内在链接$K_6美元:对于任何整数$1,2,3,4,5,6美元在3个空格中为6点1,2,4,5,6美元,其中每2美元加一条多边形线,j美元加一条多边形线,1条多边形线的内部与任何其他多边形线脱钩,任何3周期脱节的对子的连结系数除$123,456美元外为零,特殊对子的连结系数为$123,456美元为2z+1美元。