This cubic is one of my favourite curves. It has equation y2=x3. If you draw a picture
you'll
see a sharp point at the origin where there isn't a well-defined
tangent. This is a singular point of the curve.
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In fact this singularity is what stops the humble cusp from being a mighty elliptic curve. There is a bijective smoothing parametrisation of the curve given
by t --> (t2,t3).
If you intersect the complex points of the cusp
with a small sphere |x|2 + |y|2=e2, for a small positive real
number e then with some calculation you can show that
this intersection is topologically the same as the trefoil knot.
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