Torus knot

In knot theory, a torus knot is a special kind of knot which 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". The ("p","q")-torus knot winds "q" times around a circle inside the torus, which goes all the way around the torus, and "p" times around a line through the hole in the torus, which passes once through the hole, (usually drawn as an axis of symmetry). If "p" and "q" are not relatively prime, then we have a torus link with more than one component.

The ("p","q")-torus knot can be given by the parameterization:$x\; =\; left(2+cosleft(frac\{qphi\}\{p\}\; ight)\; ight)cosphi$:$y\; =\; left(2+cosleft(frac\{qphi\}\{p\}\; ight)\; ight)sinphi$:$z\; =\; sinleft(frac\{qphi\}\{p\}\; ight)$This lies on the surface of the torus given by $(r-2)^2\; +\; z^2\; =\; 1$ (in cylindrical coordinates).

Torus knots are trivial iff either "p" or "q" is equal to 1. The simplest nontrivial example is the (2,3)-torus knot, also known as the trefoil knot.

Properties

Each torus knot is prime and chiral. Any ("p","q")-torus knot can be made from a closed braid with "p" strands. The appropriate braid word is:$(sigma\_1sigma\_2cdotssigma\_\{p-1\})^q.$The crossing number of a torus knot is given by:"c" = min(("p"−1)"q", ("q"−1)"p").The genus of a torus knot is:$g\; =\; frac\{1\}\{2\}(p-1)(q-1).$The Alexander polynomial of a torus knot is:$frac\{(t^\{pq\}-1)(t-1)\}\{(t^p-1)(t^q-1)\}$The Jones polynomial of a (right-handed) torus knot is given by:$t^\{(p-1)(q-1)/2\}frac\{1-t^\{p+1\}-t^\{q+1\}+t^\{p+q\{1-t^2\}.$

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 its interior, and "X" be the quotient space obtained by identifying "Y" and "Z" along their boundary circle. The knot complement of the ("p", "q")-torus knot deformation retracts to the space "X". Therefore, the knot group of a torus knot has the presentation

:$langle\; x,y\; mid\; x^p\; =\; y^q\; angle.$

Torus knots are the only knots whose knot groups have non-trivial center (which is infinite cyclic, generated by the element $x^p\; =\; y^q$ in the presentation above).

ee also

*Alternating knot
*Cinquefoil knot
*Prime knot
*Trefoil knot

External links

*MathWorld|urlname=TorusKnot|title=Torus Knot