A thrackle is a drawing on the plane consisting of vertices (points) and edges (non-self-intersecting curves) such that:

• every edge ends in two different vertices, but hits no other vertex; and
• every edge intersects each other edge exactly once, either at a vertex or by crossing at an interior point.

Is there a thrackle with more edges than vertices?

P43846 I don't know

I don't think so, but there are with as many edges as vertices.

P43883
How many different such figures exist, for a given number of vertices? I think the first terms are
1 0
2 0
3 1
4 2
5 10, but I may have missed some configurations
I doodled them on paper, but I can't be arsed to redo them on the computer rn

P43887
wtf does that mean?

Given Conway's involvement, I wonder if this is related to some sort of game.

P44316
Why would it be? Conway worked on many things that were unrelated to any game

P43894
I'm trying to think of how to generate them in a manner that ensures we have everything. We could try ignoring the intersections first, then deciding which edges we're going to make intersect each other. Is there a way to generate all the graphs with V vertices and E edges that aren't isomorphic to each other? When I try to generate them I get a lot of duplicates due to isomorphic graphs, so I wonder if there's a more efficient way.