# 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.

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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.

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