Weakly linked embeddings of pairs of complete graphs in $\mathbb{R}^3$
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2020
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| _version_ | 1866910535445905408 |
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| author | Di, James Flapan, Erica Johnson, Spencer Thompson, Daniel Tuffley, Christopher |
| author_facet | Di, James Flapan, Erica Johnson, Spencer Thompson, Daniel Tuffley, Christopher |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2012_11030 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Weakly linked embeddings of pairs of complete graphs in $\mathbb{R}^3$ Di, James Flapan, Erica Johnson, Spencer Thompson, Daniel Tuffley, Christopher Geometric Topology 57M15, 57K10 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. |
| title | Weakly linked embeddings of pairs of complete graphs in $\mathbb{R}^3$ |
| topic | Geometric Topology 57M15, 57K10 |
| url | https://arxiv.org/abs/2012.11030 |