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.

在 纽结理论 中， 环面纽结 （torus knot）是一种特殊的结。 它由一对整参数 p 和 q 决定。 (p,q)-环面纽结可以表示为： 这个纽结所处的平面为 (r − 2) 2 + z 2 = 1（以 圓柱坐標系 表示）。 环面纽结的交叉数： c = min ( (p −1) q, (q −1) p). 种类数： 右手侧镜像的衍生数： / . {\displaystyle t^ { (p-1) (q-1)/2} {\frac {1-t^ {p+1}-t^ {q+1}+t^ {p+q}} {1-t^ {2}}}.} 结组数： , ∣ {\displaystyle \langle x,y\mid x^ {p}=y^ {q}\rangle .}

2025年3月5日 · A -torus knot is obtained by looping a string through the hole of a torus times with revolutions before joining its ends, where and are relatively prime. A -torus knot is equivalent to a -torus knot. All torus knots are prime (Hoste et al. 1998, Burde and Zieschang 2002).

2022年4月18日 · Torus knots are special types of knots which wind around a torus a number of times in the longitudinal and meridional directions. We compute and describe the fundamental group of torus knots by using some concepts in algebraic topology and group theory.

2024年6月1日 · The torus knot lies on the surface of the unknotted torus $ ( r - 2) ^ {2} + z ^ {2} = 1 $, intersecting the meridians of the torus at $ p $ points and the parallels at $ q $ points. The torus knots of types $ ( p, 1) $ and $ ( 1, q) $ are trivial. The simplest non-trivial torus knot is the trefoil (Fig. a), which is of type $ ( 2, 3) $.

knot is defined as a closed curve embedded in three-dimensional space or more simply, a string whose strands are twisted and crossed in any desired fashion and closed through connection.

One of the more important types of knot is that of the torus knot, which is any knot that is embedded onto a standard torus (one which, when solidi ed, deformation retracts onto an unknot) in S3. Typical torus knots can be expressed as follows:

One group of knots which have already been completely classified are called torus knots. As the terminology suggests, a knot is a torus knot if it is equivalent to a knot that can be drawn without any points of intersection on the trivial torus.

A trefoil is just on example of a torus knot. It is a (2,3) torus knot when means it winds 3 times around a circle in the interior of the torus, and 2 times around the torus’ axis of rotational symmetry. There is a whole family of torus knots (p,q). p and q correspond to pp and qq in the parametric formula above. For instance, here is a (3,7 ...

A special kind of knots which are always fibred are called torus knots, which are embedded in the surface of a torus. In the following visualizations we consider the trefoil knot or the (2,3)-torus knot.