# Knots and Singularities

These next few posts are about some cool things I found a book by Frances Kirwan (Complex Algebraic Curves)

### Singularities and Knots

This comes roughly from page 11 of the book. So is a complex algebraic curve, say sitting in . Look at an affine patch; that is, the zero locus of some in .

Say has a singularity at . Look at a small -sphere near the origin: for small . It turns out the intersection is a knot. Forget proving this, but consider how to extract the knot from equation .

Here are the steps involved. We want to get as a point set in . So first defined the (stereo graphic projection) .

The projection is from the point . Meaning this point maps to . For all other points let be the real line connecting and . It intersects in a unique point. Its when , or . Then the explicit formula is (for ):

Now about getting the inverse to this map. Say $(u,v,w) \in \mathbb{R}^3$. We want to produce two complex numbers. Based on the forwards map, the first should be a (real) scalar multiple of $u + iv$ and the second be roughly $x + iw$. All in all we can write the point as

you have two conditions: the value of above, and also that this point should have norm , these in principle should determine but the equations seem to get pretty ugly (the answer is in the book).

A concrete example $f = xy – y^2 = y(x – y)$. Then it is the union of the horizontal line given by and the diagonal line given by . Its not hard to check that under the sterographic projection, the horizontal line maps to a circle

and the diagonal line maps to the ellipse

and it happens that these are closed paths in that are linked. You could do this with and you would get the trefoil wrapped around the torus; you get the torus knot but the equations are more cumbersome. I wonder if you would get the torus knot if you look at ?

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- Published:
- December 11, 2009 / 7:30 pm

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- wall scribble

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