Toric knot

(3,7) -toric knot.

EureleA prize in the form of a (2,3) -toric knot.

(2.8) -toric gearing

Toric knot - a special kind of knots lying on the surface of an unknotted torus in $\mathbb {R} ^{3}$ ${\ displaystyle \ mathbb {R} ^ {3}}$ $\ mathbb {R} ^ {3}$ .

Toric gearing - gearing lying on the surface of a torus. Each toric knot is defined by a pair of mutually prime integers $p$ ${\ displaystyle p}$ $p$ and $q$ ${\ displaystyle q}$ $q$ . Toric gearing occurs when $p$ ${\ displaystyle p}$ $p$ and $q$ ${\ displaystyle q}$ $q$ are not coprime (in this case, the number of components is equal to the largest common factor $p$ ${\ displaystyle p}$ $p$ and $q$ ${\ displaystyle q}$ $q$ ) A torus knot is trivial if and only if either $p$ ${\ displaystyle p}$ $p$ either $q$ ${\ displaystyle q}$ $q$ equal to 1 or −1. The simplest non-trivial example is the (2,3) -toric knot, also known as the trefoil .

(2, −3) -toric knot, also known as left trefoil

Content

Geometric View

A toric knot can be represented in geometrically different ways, topologically equivalent, but geometrically different.

A commonly used convention is that $(p,q)$ ${\ displaystyle (p, q)}$ Torus knot rotates $q$ ${\ displaystyle q}$ times around the circular axis of the torus and $p$ ${\ displaystyle p}$ times around the axis of rotation of the torus. If a $p$ ${\ displaystyle p}$ and $q$ ${\ displaystyle q}$ are not mutually simple, then we obtain a toric gearing having more than one component. Agreements on the direction in which the threads rotate around the torus are also different, most often the right screw for $pq>0$ ${\ displaystyle pq> 0}$ ^[1] ^[2] ^[3] .

$(p,q)$ ${\ displaystyle (p, q)}$ torus node can be specified by :

x=r

{\ displaystyle x = r}

\cos(p\phi )

{\ displaystyle \ cos (p \ phi)}

,

y=r

{\ displaystyle y = r}

\sin(p\phi )

{\ displaystyle \ sin (p \ phi)}

,

z=-\sin(q\phi )

{\ displaystyle z = - \ sin (q \ phi)}

,

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

Other parameterizations are also possible, since the nodes are determined up to a continuous deformation. Examples for (2,3) - and (3,8) -toric knots can be obtained by accepting $r=\cos(q\phi )+4$ ${\ displaystyle r = \ cos (q \ phi) +4}$ , and in the case of a (2,3) -toric knot by subtraction $3\cos((p-q)\phi )$ ${\ displaystyle 3 \ cos ((pq) \ phi)}$ and $3\sin((p-q)\phi )$ ${\ displaystyle 3 \ sin ((pq) \ phi)}$ from the above parameterizations $x$ ${\ displaystyle x}$ and $y$ ${\ displaystyle y}$ .

Properties

Diagram of a (3, −8) -toric knot.

A torus knot is trivial if and only if either $p$ ${\ displaystyle p}$ either $q$ ${\ displaystyle q}$ equal to 1 or −1 ^[2] ^[3] .

Each nontrivial toric knot is simple and chiral .

$(p,q)$ ${\ displaystyle (p, q)}$ the torus knot is equivalent $(q,p)$ ${\ displaystyle (q, p)}$ -toric node ^[1] ^[3] . $(p,-q)$ ${\ displaystyle (p, -q)}$ torus node is the inverse (mirror reflection) $(p,q)$ ${\ displaystyle (p, q)}$ torus node ^[3] . $(-p,-q)$ ${\ displaystyle (-p, -q)}$ the torus knot is equivalent $(p,q)$ ${\ displaystyle (p, q)}$ Torus node, except for orientation.

(3, 4) a toric knot on a turn of the surface of a torus and the word braids

Any $(p,q)$ ${\ displaystyle (p, q)}$ torus node can be constructed from a closed braid with $p$ ${\ displaystyle p}$ threads. Suitable braid word ^[4] :

(\sigma _{1}\sigma _{2}\cdots \sigma _{p-1})^{q}

{\ displaystyle (\ sigma _ {1} \ sigma _ {2} \ cdots \ sigma _ {p-1}) ^ {q}}

.

This formula uses the convention that braid generators use right-handed rotations ^[2] ^[4] ^[5] ^[6] .

Number of intersections $(p,q)$ ${\ displaystyle (p, q)}$ Torus node with $p,q>0$ ${\ displaystyle p, q> 0}$ is given by the formula:

c=\min((p-1)q,(q-1)p)

{\ displaystyle c = \ min ((p-1) q, (q-1) p)}

.

The genus of a toric knot with $p,q>0$ ${\ displaystyle p, q> 0}$ is equal to:

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

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

The Alexander polynomial of the torus knot is ^[1] ^[4] :

{\frac {(t^{pq}-1)(t-1)}{(t^{p}-1)(t^{q}-1)}}

{\ displaystyle {\ frac {(t ^ {pq} -1) (t-1)} {(t ^ {p} -1) (t ^ {q} -1)}}}

.

The Jones polynomial (right-handed) of a toric knot is given by the formula:

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

{\ 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 toric knot on a 3-sphere is a Seifert manifold .

Let be $Y$ ${\ displaystyle Y}$ - $p$ ${\ displaystyle p}$ -dimensional with a disk removed inside, $Z$ ${\ displaystyle Z}$ - $q$ ${\ displaystyle q}$ a stupid dimensional hood with an internal remote drive, and $X$ ${\ displaystyle X}$ - factor space obtained by identification $Y$ ${\ displaystyle Y}$ and $Z$ ${\ displaystyle Z}$ along the boundary of the circle. Addition $(p,q)$ ${\ displaystyle (p, q)}$ - toric knot is a deformation retract of space $X$ ${\ displaystyle X}$ . Thus, the torus knot node group has the representation :

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

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

.

Toric knots are the only knots whose knot groups have nontrivial centers (which are infinite cyclic groups formed by an element $x^{p}=y^{q}$ ${\ displaystyle x ^ {p} = y ^ {q}}$ from this view).

List

Trivial knot , 3 _1- knot (2,3), Knot "Noblet" (5,2), (7,2), 8 _19- knot (4,3), 9 _1- knot ( 9.2), 10 ₁₂₄ -unit (5.3).

Notes

↑ ¹ ² ³ Livingston, 1993 .
↑ ¹ ² ³ Murasugi, 1996 .
↑ ¹ ² ³ ⁴ Kawauchi, 1996 .
↑ ¹ ² ³ Lickorish, 1997 .
↑ Dehornoy, P. et al. (2000). Why are braids orderable? http://www.math.unicaen.fr/~dehornoy/Books/Why/Dgr.pdf Archived April 15, 2012 on the Wayback Machine
↑ Birman, Brendle, 2005 .

Literature

Charles Livingston Knot theory. - Mathematical Association of America, 1993. - ISBN 0-88385-027-3 .
Kunio Murasugi. Knot theory and its applications. - Birkhäuser, 1996. - ISBN 3-7643-3817-2 .
Akio Kawauchi. A survey of knot theory. - Birkhäuser, 1996. - ISBN 3-7643-5124-1 .
WBR Lickorish. An introduction to knot theory. - Springer, 1997 .-- ISBN 0-387-98254-X .
JS Birman, TE Brendle. Handbook of knot theory / W. Menasco, M. Thistlethwaite. - Elsevier, 2005. - ISBN 0-444-51452-X ..
J. Milnor. Singular Points of Complex Hypersurfaces. - Princeton University Press, 1968. - ISBN 0-691-08065-8 .

Links

36 Torus Knots , The Knot Atlas.
Weisstein, Eric W. Torus Knot ( Wolfram ) at Wolfram MathWorld .
Torus knot renderer in Actionscript
Fun with the PQ-Torus Knot

[_a63f86a9533c0d18-1] ¹ ² ³ Livingston, 1993 .

[_4032bf0da1fcc115-2] ¹ ² ³ Murasugi, 1996 .

[_c51bbd8211bbd8dd-3] ¹ ² ³ ⁴ Kawauchi, 1996 .

[_f5d8b1d87958ebd9-4] ¹ ² ³ Lickorish, 1997 .

[dehornoy-5] Dehornoy, P. et al. (2000). Why are braids orderable? http://www.math.unicaen.fr/~dehornoy/Books/Why/Dgr.pdf Archived April 15, 2012 on the Wayback Machine

[_c74856210a9fa53b-6] Birman, Brendle, 2005 .