Torus knot

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. 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. A torus knot can be rendered geometrically in multiple ways which are topologically equivalent (see Properties below) but geometrically distinct. The convention used in this article and its figures is the following. 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. {Note, this use of the roles of p and q is contrary to what appears on: http://mathworld.wolfram.com/TorusKnot.htmlIt is also inconsistent with the 'List' of torus knots below and with the pictures that appear in: '36 Torus Knots', The Knot Atlas.}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. The (p,q)-torus knot can be given by the parametrization where r = cos ⁡ ( q ϕ ) + 2 {displaystyle r=cos(qphi )+2} and 0 < ϕ < 2 π {displaystyle 0<phi <2pi } . This lies on the surface of the torus given by ( r − 2 ) 2 + z 2 = 1 {displaystyle (r-2)^{2}+z^{2}=1} (in cylindrical coordinates). 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 r = cos ⁡ ( q ϕ ) + 4 {displaystyle r=cos(qphi )+4} , and in the case of the (2,3)-torus knot by furthermore subtracting respectively 3 cos ⁡ ( ( p − q ) ϕ ) {displaystyle 3cos((p-q)phi )} and 3 sin ⁡ ( ( p − q ) ϕ ) {displaystyle 3sin((p-q)phi )} from the above parameterizations of x and y. The latter generalizes smoothly to any coprime p,q satisfying p < q < 2 p {displaystyle p<q<2p} . A torus knot is trivial iff either p or q is equal to 1 or −1. Each nontrivial torus knot is prime and chiral.