Torus knot
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.
A torus knot can be rendered geometrically in multiple ways which are topologically equivalent (see Properties below) but geometrically distinct.
The (p,q)-torus knot winds q times around a circle in the interior of the torus, and p times around its axis of rotational symmetry.
If p and q are not relatively prime, then we have a torus link with more than one component.
The direction in which the strands of the knot wrap around the torus is also subject to differing conventions.
The most common is to have the strands form a right-handed screw for p q > 0.
Other parameterizations are also possible, because knots are defined up to continuous deformation.
The illustrations for the (2,3)- and (3,8)-torus knots can be obtained by taking
The latter generalizes smoothly to any coprime p,q satisfying
A torus knot is trivial iff either p or q is equal to 1 or −1.
[4][5] Each nontrivial torus knot is prime[6] and chiral.
[3][5] This can be proved by moving the strands on the surface of the torus.
[7] The (p,−q) torus knot is the obverse (mirror image) of the (p,q) torus knot.
[5] The (−p,−q) torus knot is equivalent to the (p,q) torus knot except for the reversed orientation.
Any (p,q)-torus knot can be made from a closed braid with p strands.
The appropriate braid word is [8] (This formula assumes the common convention that braid generators are right twists,[4][8][9][10] which is not followed by the Wikipedia page on braids.)
The crossing number of a (p,q) torus knot with p,q > 0 is given by The genus of a torus knot with p,q > 0 is The Alexander polynomial of a torus knot is [3][8] The Jones polynomial of a (right-handed) torus knot is given by The complement of a torus knot in the 3-sphere is a Seifert-fibered manifold, fibred over the disc with two singular fibres.
Let Y be the p-fold dunce cap with a disk removed from the interior, Z be the q-fold dunce cap with a disk removed from its interior, and X be the quotient space obtained by identifying Y and Z along their boundary circle.
The stretch factor of the (p,q) torus knot, as a curve in Euclidean space, is Ω(min(p,q)), so torus knots have unbounded stretch factors.
Undergraduate researcher John Pardon won the 2012 Morgan Prize for his research proving this result, which solved a problem originally posed by Mikhail Gromov.
[11][12] 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.
This intersection is the so-called link of the singularity
, where p and q are coprime, and both greater than or equal to two, is exactly the (p,q)−torus knot.
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).
