Knotilus is an interactive tool for drawing and studying the various properties of knots and links.

Extensive help/information is available for every aspect of this website, simply click on the label for an element to open a new window with a documentation page.

Selecting a Knot/Link to Draw

Database of Knot/Links

From the main page, one can browse and select to draw any of the more than 98 billion prime alternating knots/links at or below 23 crossings. As well we provide a few large examples, a list of user specified drawings and some other interesting classes of knot/links.

Gauss Code/Dowker Code

You can enter the Dowker code for a knot, using spaces, or a Gauss code for a knot/or link, in the entry box. When using the automated drawing, there is a limit of 1000 characters in the code, for the interactive Java drawing tool, there is no limit to the number of crossings.

Link Sketcher

In the top menu, we link to our Java based Link Sketcher and annealing tool, you can draw a link freehand and then the applet will redraw it using our algorithm.

Various Numbering Schemes

Throughout the history of of knot enumeration there have been various systems used to identify knots, we provide the ability for one to enter the Alexander-Briggs or Thistlethwaite numbers and retrieve the knot/link. As well we give quick access using our own scheme for all prime alternating knots/links at or below 23 crossings. As well, if a prime alternating knot/link has such a number attached to it, we will display it with the drawing along with other properties.

Methods of Drawing

There are two implementations of our algorithm:

Automated Drawing

This is the default mode, the drawing is entered in a queue and the server draws the knot/link with one of the largest faces as the exterior using the default parameters. This usually produces a pleasing diagram. One can rotate, mirror and choose a different exterior face for the diagram interactively. Only a limited number of crossings are supported in order to reduce the load on the server.

Java Based Annealer

By selecting the check box beside the label 'Draw in Java annealer', you are presented with an initial planar drawing of the underlying graph and complete control of the parameters of the algorithm. You can interactively rotate, invert and alter the diagram as well.

Output Modes

We support various outputs described here.

Calculation of Properties

When possible we provide the user with a wide range of properties described here.