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.

|     .
|     
|    /
|   .
| .
|--------------------
| .
|   .
|    \
|     
|     .

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.

Log in or registerto write something here or to contact authors.