Торус торындағы түйін және сілтемелер

Knot which lies on the surface of a torus in 3 dimensional space

In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of coprime integers p and q. A torus link arises if p and q are not coprime (in which case the number of components is gcd(p, q)). A torus knot is trivial (equivalent to the unknot) if and only if either p or q is equal to 1 or −1. The simplest nontrivial example is the (2,3) torus knot, also known as the trefoil knot.

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 The 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.

A g torus knot is a closed curve drawn on a g torus. More technically, it is the homeomorphic image of a circle in S³ which can be realized as a subset of a genus g handlebody in S³ (whose complement is also a genus g handlebody). If a link is a subset of a genus two handlebody, it is a double torus link. For genus two, the simplest example of a double torus knot that is not a torus knot is the figure eight knot.