The KnotPlot Site

Welcome to the KnotPlot Site!
Here you will find a collection of knots and links, viewed from a (mostly) mathematical perspective. Nearly all of the images here were created with KnotPlot, an elaborate program to visualize and manipulate mathematical knots in three and four dimensions.

Download KnotPlot for Windows, macOS and Linux

Knot Pictures

Check out the mathematical knots (M) page as well to see more knot pictures. Or try some of the following examples to see some knots in a different light. The pages marked with

have been updated or created as of 8 December 2020.

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Various Picture Galleries

Knots as radiating tubes M
More knots as radiating tubes
Decorative knots: on white, on black, goth
Antoine's Necklace (a fractal link)
Moiré pictures — not knot like, but still rendered with KnotPlot
Ashley knots
Nifty Knots with Fancy Lighting
A knot zoo (interactive 3D viewer with VR option)
Torus knots and links, arranged by crossing number
Composite Links
Brunnian Links M
Linking Spheres in three or more dimensions
PDF and PostScript examples
Knot Carpets
3D Stereoscopic Pictures (page 1, page 2, page 3, page 4, page 5, page 6)
Celtic Knots
Hyperbolic knots (interactive 3D viewer with VR option)
Complex knots
Perko pair knots
Moebius strip knots
Coney
Project Aum
Stick numbers for minimal stick knots (data and interactive 3D viewer)
Some excellent references on knot theory.
Some favourite figures from my thesis (PDF and PostScript)
Symmetric projections of minimal-stick knots
Links to other peoples' knot sites.

Raytraced Examples

Other pages

Quote from Clifford Ashley
Interactive Topological Drawing, my dissertation
Short discussion on braid theory (incomplete).

Pages with little or no interesting content

This section contains pages that are badly in need of updating! Or possibly deletion.

Knotted Sea Things
Various Odd Knots, Unknot Flowers M
Knot Diagrams from Dowker Codes (better information is available in my dissertation)
Knots on the Cubic Lattice
Hopf Torus M

KnotPlot

KnotPlot features

Knots can be loaded from a database of more than 3,000 knots and links or sketched by hand in three dimensions. Also, knots may be constructed via the Conway notation or using the tangle calculator. A number of special knot types (torus knots, knot chains, Lissajous knots) may be created on the fly. Finally, new knots can be created from old knots by various tranformation tools. Using all these features allows the creation of quadrillions of different knots (and many more).

Downloading KnotPlot

KnotPlot is an OpenGL program that runs on most types of computers. Go to the download site for a copy.

Related work

Check out these links to see related work by other people.

Author

Robert Scharein,< @hypnagogic.net>
If you have comments on these pages please feel free to email them to me.

KnotPlot was part of my PhD thesis work in the Imager Computer Graphics Laboratory. The warmest appreciation goes to my supervisory committee Dr. Kellogg S. Booth, Dr. Jim Little, Dr. Nicholas Pippenger and Dr. Dale Rolfsen.

KnotPlot is provided by Hypnagogic Software.

Acknowledgements

A big thank you to Dale Rolfsen, Jonathan Simon, and Greg Buck for much inspiration and support, especially for getting me started on the KnotPlot project.

Initial KnotPlot development was funded in part by the Department of Computer Science at the University of British Columbia (UBC/CS), the Media and Graphics Interdisciplinary Centre (MAGIC), the Pacific Institute for the Mathematical Sciences (PIMS), and the Centre for Experimental and Constructive Mathematics (CECM). The hugest of hugs to PIMS, CECM, and UBC/CS for hosting the KnotPlot pages over all these years.

no symbols where none intended