Torus knot

2020 Mathematics Subject Classification: Primary: 57K [MSN][ZBL]

of type $ ( p, q) $

A curve in $ \mathbf R ^ {3} $ that in cylindrical coordinates $ r, z, \theta $ is given by the equations

$$ r = 2 + \cos t,\ \ z = \sin t,\ \ \theta = { \frac{pt }{q} } , $$

where $ t \in [ 0, 2 \pi q] $. Here $ p $ and $ q $ are coprime natural numbers. 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) $. The group of the torus knot of type $ ( p, q) $ has a presentation $ < a, b $: $ a ^ {p} = b ^ {q} > $, and the Alexander polynomial is given by

$$ ( t ^ {pq} - 1) ( t - 1) ( t ^ {p} - 1) ^ {-} 1 ( t ^ {q} - 1) ^ {-} 1 . $$

All torus knots are Neuwirth knots (cf. Neuwirth knot). The genus of a torus knot is $ ( p - 1) ( q - 1)/2 $.

A second construction of a torus knot uses the singularity at the origin of the algebraic hypersurface

$$ V = \ \{ {( z _ {1} , z _ {2} ) \in \mathbf C ^ {2} } : { z _ {1} ^ {p} + z _ {2} ^ {q} = 0 } \} . $$

If $ p $ and $ q $ are coprime, then the intersection of $ V $ with a sufficiently small sphere $ S ^ {3} \subset \mathbf C ^ {2} $ is a knot in $ S ^ {3} $ equivalent to the torus knot of type $ ( p, q) $. In the case when $ p $ and $ q $ are not coprime, this intersection also lies on an unknotted torus $ T ^ {2} \subset S ^ {3} $, but consists of several components. The link so obtained is called the torus link of type $ ( p, q) $( cf. Fig. b, where $ p = 3 $, $ q = 6 $).

Figure: t093360a

Figure: t093360b

Comments

See also Knot theory.

References

[1]	R.H. Crowell, R.H. Fox, "Introduction to knot theory" , Ginn (1963)
[2]	J. Milnor, "Singular points of complex hypersurfaces" , Princeton Univ. Press (1968)
[a1]	D. Rolfsen, "Knots and links" , Publish or Perish (1976)