论文标题
$ \ mathbb {r}^3 $中的一对完整图的弱链接嵌入
Weakly linked embeddings of pairs of complete graphs in $\mathbb{R}^3$
论文作者
论文摘要
令$ g $和$ h $是完整图的嵌入$ k_m $和$ k_n $ in $ \ mathbb {r}^3 $中的$ k_n $，以至于$ g $中的某个周期链接在$ h $中的$ h $中，具有非零链接数。我们说，如果$ g $和$ h $ *弱链接 *，如果$ g $中任何周期的链接数的绝对值，$ h $中的周期为$ 0 $或$ 1 $。我们的主要结果是一对脱节的完整图被微弱链接时的代数表征。 作为朝着此结果迈出的一步，我们表明，如果$ g $和$ h $链接较弱，那么每个都包含一个链接对方的所有三角形的顶点，或者包含一个与所有三角形链接的三角形的三角形。然后，所有弱连接的完整图对的家族都以每个完整图中的两个情况中的哪个表征来表征。
Let $G$ and $H$ be disjoint embeddings of complete graphs $K_m$ and $K_n$ in $\mathbb{R}^3$ such that some cycle in $G$ links a cycle in $H$ with non-zero linking number. We say that $G$ and $H$ are *weakly linked* if the absolute value of the linking number of any cycle in $G$ with a cycle in $H$ is $0$ or $1$. Our main result is an algebraic characterisation of when a pair of disjointly embedded complete graphs is weakly linked. As a step towards this result, we show that if $G$ and $H$ are weakly linked, then each contains either a vertex common to all triangles linking the other or a triangle which shares an edge with all triangles linking the other. All families of weakly linked pairs of complete graphs are then characterised by which of these two cases holds in each complete graph.
