Geodesic
Weakly linked embeddings of pairs of complete graphs in \(\mathbb{R}^3\)
James Di1 ; Erica Flapan2 ; Spencer Johnson3 ; Daniel Thompson4 ; Christopher Tuffley5
1ByteDance
2Department of Mathematics, Pomona College
3Austin Community College
4Washington Post
5School of Fundamental Sciences, Massey University
The electronic journal of combinatorics, Tome 29 (2022) no. 2
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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 embedded complete graphs are then characterised by which of these two cases holds in each complete graph.
DOI : 10.37236/10322
Classification : 57M15, 57K10
Mots-clés : intrinsically linked, spatial graph

James Di  1   ; Erica Flapan  2   ; Spencer Johnson  3   ; Daniel Thompson  4   ; Christopher Tuffley  5

1 ByteDance
2 Department of Mathematics, Pomona College
3 Austin Community College
4 Washington Post
5 School of Fundamental Sciences, Massey University 
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     title = {Weakly linked embeddings of pairs of complete graphs in {\(\mathbb{R}^3\)}},
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James Di; Erica Flapan; Spencer Johnson; Daniel Thompson; Christopher Tuffley. Weakly linked embeddings of pairs of complete graphs in \(\mathbb{R}^3\). The electronic journal of combinatorics, Tome 29 (2022) no. 2. doi: 10.37236/10322

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