In this help document, we describe the five main windows in the upper half of the knotilus home page. Their combined role is to allow the user to select a knot or link from either the archive of all prime alternating links up to 23 crossings or from either the list of examples or the list of user-entered knots or links.
To choose a knot or link from the archive, select the desired number of crossings from the Crossing list, then the number of components from the Component list and finally, choose the desired type from the Class list. For a detailed description of the various types available, please see the Preprint page for Stuart Rankin.
When the desired category has been selected, the knots/links in that category are listed in the window entitled Gauss/Dowker codes, in sets of 64 if there are more than 64 knots/links in the category. The total number of sets is displayed below the set selection text box. You may use the increase/decrease buttons to modify the number of the desired set, or (when the number of sets is very large) you may type in the set number in the text box and then click on the Go button (or press Enter).
Once you have the code for the desired knot/link visible in the Gauss/Dowker codes window, you must double click on the code in order to have it drawn. The drawing will be displayed in a separate window.
As for the apparent choice of Gauss versus Dowker code, one should note that the only time that a Dowker code will be available for selection is if someone has previously entered a Dowker code manually, in which case the Dowker code will be displayed in the User Specified list and is thereby available for selection. All knots or links from the archives will be presented by their number in the archive, followed by a gauss code (what we call the 0-position gauss code for the knot or link).
It should be noted that if you already have the archive number for a knot or link from our archive, or if you have either the Alexander-Briggs or the Thistlethwaite number for an alternating knot or link, you may enter that in the appropriate text box in the lower half of the Knotilus web page. Click here for a description of these different schemes.
The form of the archive number is <integer1>x-<integer2>-<integer3>, where the first integer is the crossing size, the second the number of components, while the last integer is the number of the knot or link within that collection. For example, 23x-2-10020 is the archive number of the 10,020th 2-component prime alternating link of 23 crossings.