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Torus Knots - Definitions and Properties

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Defining excerpt from Quantum Mechanics of Particle on a torus knot: Curvature and Torsion Effects
   —A PREPRINT

A torus is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution.[1]
Torus knots are a special class of non-self-intersecting, closed curves, wound on the surface of a geometric torus in R3. The family of knots appear because the first homotopy group for the torus is π1(torus) ≈ Z1 × Z1.
For our purposes, each knot is defined using 2 co-prime integers, p, q ∈ N. A (p, q)−torus knot winds p times in the toroidal direction, about the axis of rotational symmetry of the torus and q times in the poloidal direction, about the cross-sectional axis of symmetry (see Fig.1).
The quantity α = q/p is called the ’winding number’ of the (p, q)− torus knot, and is a simple measure of the complexity of the knot. One may note immediately, that two different torus knots (differing in at least one of p or q), will have different values of α. In this sense, α, may be regarded as the ’unique’ identity of a (p, q)−torus knot. Some popular examples of torus knots are depicted in Fig. 2, below.
Figure 1: (Left) Red arrow depicts poloidal direction, while blue arrow depicts toroidal direction. (Right) The major and minor radii of a torus (Picture Courtesy - P. Das et. al. [49]) .
File:Quantum mechanics on a torus knot fig.2 572x224.png
Figure 2: (Left) A trefoil knot, or a (2, 3)−torus knot, is the simplest non-trivial knot. (Right) A (3,10)-torus knot.
  1. As the distance from the axis of revolution decreases, the ring torus becomes a horn torus, then a spindle torus, and finally degenerates into a sphere.