This
cubic is one of my favourite
curves. It has
equation y^{2}=x^{3}. If you draw a picture
you'll
see a sharp point at the
origin where there isn't a welldefined
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 > (t^{2},t^{3}).
If you intersect the complex points of the cusp
with a small sphere x^{2} + y^{2}=e^{2}, 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.