Abstract
We investigate the minimal number of links and knots in embeddings of complete partite graphs in $S^3$. We provide exact values or bounds on the minimal number of links for all complete partite graphs with all but 4 vertices in one partition, or with 9 vertices in total. In particular, we find that the minimal number of links in an embedding of $K_{4,4,1}$ is 74. We also provide exact values or bounds on the minimal number of knots for all complete partite graphs with 8 vertices.
Citation
Loren ABRAMS. Blake MELLOR. Lowell TROTT. "Counting Links and Knots in Complete Graphs." Tokyo J. Math. 36 (2) 429 - 458, December 2013. https://doi.org/10.3836/tjm/1391177980
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