Toroidal Complete Graphs

In the graphs above, only \(K_4\) can be drawn on a plane without any edge crossings, as illustrated below:

This cannot be done with \(K_5\), yet it can be drawn on a torus without any edge crossings:

Top view of the torus. The blue dotted line is on the bottom side.

Top view of the torus. The blue dotted line is on the bottom side.

Can \(K_6\) and \(K_7\) be drawn on a torus without edge crossings?

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