Torus knot

In

Geometrical representation

A torus knot can be rendered geometrically in multiple ways which are

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 1]}. 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.^[3][4][5]

The (p,q)-torus knot can be given by the parametrization

${\displaystyle {\begin{aligned}x&=r\cos(p\phi )\\y&=r\sin(p\phi )\\z&=-\sin(q\phi )\end{aligned}}}$

where ${\displaystyle r=\cos(q\phi )+2}$ and ${\displaystyle 0<\phi <2\pi }$. This lies on the surface of the torus given by ${\displaystyle (r-2)^{2}+z^{2}=1}$ (in

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 ${\displaystyle r=\cos(q\phi )+4}$, and in the case of the (2,3)-torus knot by furthermore subtracting respectively ${\displaystyle 3\cos((p-q)\phi )}$ and ${\displaystyle 3\sin((p-q)\phi )}$ from the above parameterizations of x and y. The latter generalizes smoothly to any coprime p,q satisfying ${\displaystyle p<q<2p}$.

Properties

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.^[4]

The (p,q) torus knot is equivalent to the (q,p) torus knot.^[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

(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

c = min((p−1)q, (q−1)p).

The

${\displaystyle g={\frac {1}{2}}(p-1)(q-1).}$

The Alexander polynomial of a torus knot is ^[3][8]

${\displaystyle t^{k}{\frac {(t^{pq}-1)(t-1)}{(t^{p}-1)(t^{q}-1)}},}$ where ${\displaystyle k=-{\frac {(p-1)(q-1)}{2}}.}$

The Jones polynomial of a (right-handed) torus knot is given by

${\displaystyle t^{(p-1)(q-1)/2}{\frac {1-t^{p+1}-t^{q+1}+t^{p+q}}{1-t^{2}}}.}$

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

${\displaystyle \langle x,y\mid x^{p}=y^{q}\rangle .}$

Torus knots are the only knots whose knot groups have nontrivial

center

(which is infinite cyclic, generated by the element ${\displaystyle x^{p}=y^{q}}$ in the presentation above).

The

Let p and q be coprime integers, greater than or equal to two. Consider the holomorphic function ${\displaystyle f:\mathbb {C} ^{2}\to \mathbb {C} }$ given by ${\displaystyle f(w,z):=w^{p}+z^{q}.}$ Let ${\displaystyle V_{f}\subset \mathbb {C} ^{2}}$ be the set of ${\displaystyle (w,z)\in \mathbb {C} ^{2}}$ such that ${\displaystyle f(w,z)=0.}$ Given a real number ${\displaystyle 0<\varepsilon \ll 1,}$ we define the real three-sphere ${\displaystyle \mathbb {S} _{\varepsilon }^{3}\subset \mathbb {R} ^{4}\hookrightarrow \mathbb {C} ^{2}}$ as given by ${\displaystyle |w|^{2}+|z|^{2}=\varepsilon ^{2}.}$ The function ${\displaystyle f}$ has an isolated critical point at ${\displaystyle (0,0)\in \mathbb {C} ^{2}}$ since ${\displaystyle \partial f/\partial w=\partial f/\partial z=0}$ if and only if ${\displaystyle w=z=0.}$ Thus, we consider the structure of ${\displaystyle V_{f}}$ close to ${\displaystyle (0,0)\in \mathbb {C} ^{2}.}$ In order to do this, we consider the intersection ${\displaystyle V_{f}\cap \mathbb {S} _{\varepsilon }^{3}\subset \mathbb {S} _{\varepsilon }^{3}.}$ This intersection is the so-called link of the singularity ${\displaystyle f(w,z)=w^{p}+z^{q}.}$ The link of ${\displaystyle f(w,z)=w^{p}+z^{q}}$, where p and q are coprime, and both greater than or equal to two, is exactly the (p,q)−torus knot.^[13]

List

The figure on the right is torus link (72,4) .

• 7₁ knot
  (7,2), 8₁₉ knot (4,3), 9₁ knot (9,2), 10₁₂₄ knot (5,3)

Table #	A-B	Image	P	Q	Cross #
0	01				0
3a1	31		2	3	3
5a2	51		2	5	5
7a7	71		2	7	7
8n3	819		3	4	8
9a41	91		2	9	9
10n21	10124		3	5	10
11a367			2	11	11
13a4878			2	13	13
14n21881			3	7	14
15n41185			4	5	15
15a85263			2	15	15
16n783154			3	8	16
			2	17	17
			2	19	19
			3	10	20
			4	7	21
			2	21	21
			3	11	22
			2	23	23
			5	6	24
			2	25	25
			3	13	26
			4	9	27
			2	27	27
			5	7	28
			3	14	28
			2	29	29
			2	31	31
			5	8	32
			3	16	32
			4	11	33
			2	33	33
			3	17	34
			6	7	35
			2	35	35
			5	9	36
			7	8	48
			7	9	54
			8	9	63

g-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.^[14]

For genus two, the simplest example of a double torus knot that is not a torus knot is the figure-eight knot.^[15][16]

Notes

See also

• Alternating knot
• Satellite knot
• Hyperbolic knot
• Irrational winding of a torus
• Topopolis

References