Connection To Complex Hypersurfaces
The (p,q)−torus knots arise when considering the link of an isolated complex hypersurface singularity. One intersects the complex hypersurface with a hypersphere, centred at the isolated singular point, and with sufficiently small radius so that it does not enclose, nor encounter, any other singular points. The intersection gives a submanifold of the hypersphere.
Let p and q be coprime integers, greater than or equal to two. Consider the holomorphic function given by Let be the set of such that Given a real number we define the real three-sphere as given by Tne function has an isolated critical point at since if and only if Thus, we consider the structure of close to In order to do this, we consider the intersection This intersection is the so-called link of the singularity The link of, where p and q are coprime, and both greater than or equal to two, is exactly the (p,q)−torus knot.
Read more about this topic: Torus Knot
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