This is topologically equivalent to a torus, and you can go up to 7 points:

>! [![enter image description here][1]][1]

as shown by [this Math.SE answer](https://math.stackexchange.com/a/66099/256930).

The diagram for this could look for example like this:
>! <img src="https://i.sstatic.net/qkGMn.png" title="2D representation of 7 connected points on the torus"><br>
>! In this picture the lines going "under" the square represent connections going through the cup handle and the lines going "over" the square would go along the handle of the cup.

One can also just look up the answer if you know the question asks for the <a href="https://en.wikipedia.org/wiki/Complete_graph">complete graph $K_n$ of degree $n$</a> with maximum $n$ such that the <a href="http://mathworld.wolfram.com/GraphGenus.html">graph genus $\gamma (K_n)$</a> is at most $1$. Then
>! if you take the equation from <a href="http://mathworld.wolfram.com/GraphGenus.html">Wolfram MathWorld</a>
>! $$ \gamma (K_n) = \left\lceil \frac{(n-3)(n-4)}{12} \right\rceil $$
>! you see that the genus $\gamma (K_n) \le 1$ as long as $n \le 7$.


  [1]: https://i.sstatic.net/TB9Qs.png