The Jørgensen graph is a maximally linklessly embeddable graph on 8 vertices and 21 edges, where "maximal" means it is not a proper subgraph of another linklessly embeddable graph of the same order (Jørgensen 1989, Naimi et al. 2020). It is illustrated above in a number of embeddings.
A family of maximally linklessly embeddable graphs on 
vertices and 
edges may be constructed from this graph by subdividing one of the horizontal edges
and adding edges that connect every new vertex to two the top and bottom vertices
(Jørgensen 1989, Naimi et al. 2020).
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Report 1989: R 89-28. Aalborg, Denmark: Aalborg Universitetscenter, Institut for
Elektroniske Systemer, 1989.Naimi, R.; Pavelescu, A.; and Pavelescu,
E. "New Bounds on Maximal Linkless Graphs." 20 Sep 2020.