Surfaces Defined on Torus Knots

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A torus knot is a particular type of knot that is defined on the surface of an unknotted torus. It is possible to define a surface whose edges coincide with a knot or link, which can be a non-oriented surface, as presented in this Demonstration, or an orientable surface, in which case it is named a Seifert surface, for Herbert Seifert, who introduced a general method for any knot in 1934.


The method applied here is simply starting with a one-to-one mapping between the interval and and defining lines between those points, giving a ruled surface; afterward, the relaxation method (which takes the mean of the four neighbors for each point, keeping fixed boundaries) is applied for a number of steps to give a smoother surface.


Contributed by: Enrique Zeleny (March 2015)
Open content licensed under CC BY-NC-SA



A torus knot is defined as

where is the link crossing number.


[1] J. J. van Wijk, "Seifert Surfaces," Mathematical Imagery. (Feb 11, 2015)

[2] J. J. van Wijk and A. M. Cohen, "Visualization of Seifert Surfaces," IEEE Transactions on Visualization and Computer Graphics, 1(10), 2006.

[3] The Knot Atlas. (Feb 11, 2015)

[4] A. Gray, "Torus Knots," Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed., Boca Raton, FL: CRC Press, 1997 pp. 209-215.

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