Surfaces Defined on Torus Knots

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

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.

[more]

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.

[less]

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


Snapshots


Details

A torus knot is defined as

where is the link crossing number.

References

[1] J. J. van Wijk, "Seifert Surfaces," Mathematical Imagery. (Feb 11, 2015) www.ams.org/mathimagery/thumbnails.php?album=14.

[2] J. J. van Wijk and A. M. Cohen, "Visualization of Seifert Surfaces," IEEE Transactions on Visualization and Computer Graphics, 1(10), 2006. www.maths.ed.ac.uk/~aar/papers/vanwijk.pdf.

[3] The Knot Atlas. (Feb 11, 2015) katlas.math.toronto.edu/wiki/Main_Page.

[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.



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
Send