is one of my favourite curve
s. It has equation y2=x3
. If you draw a picture
see a sharp point at the origin
where there isn't a well-defined
. This is a singular
point of the curve.
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.