The reference to colours here is inessential. Colourability is in fact a specific case of a kind of combinatorial property known as mod \(p\) labelling (for \(p\) an odd prime). Colourability is a knot invariant in the sense that if one diagram of a knot is colourable every diagram of that knot and of any equivalent knot is colourable. (By Reidemeister’s theorem this can be proved by showing that each Reidemeister move preserves colourability.) A standard diagram of an unknot, a diagram without crossings, is clearly not colourable because it has only one arc (the whole thing) and so two colours cannot be used. So in order to complete proving that the trefoil is not equivalent to an unknot, we only need prove that our trefoil diagram is colourable. This can be done visually. Colour each arc of the knot diagram one of the three colours red, green or blue so that no two arcs have the same colour (or visualize this). Then do a visual check of each crossing, to see that at each crossing the three meeting arcs are all coloured differently. That visual part of the proof is clearly non-superfluous and non-replaceable (without changing the proof). Moreover, the soundness of the argument is quite transparent.  So here is a case of a non-formal, non-symbolic visual way of proving a mathematical truth.