Link Identifiers

Knotilus Archive Number

Every prime alternating knot or link of at most 23 crossings has been assigned an archive number of the form:

<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.

Once the prime alternating links of a given crossing size and number of components have been generated, the master array is used to construct a standard gauss code for each link, which is the converted to a dowker code. Thus we have assigned a standard dowker code to each prime alternating link of the given crossing size and number of components. The collection of standard dowker codes is then sorted and then each dowker code is associated with its position in the sorted list. This position is recorded as the third integer in the description provided above.

Alexander-Briggs Notation

In the paper "On types of knotted curves", Annals of Mathematics, 28:562--586, 1926-1927, James W Alexander and G B Briggs introduced an archive numbering scheme for all of the prime knots and links of ten crossings or less. Their notation was of the form:

<<integer1>^<integer2>>_<integer3>

where integer1 is the crossing size, integer 2 is the number of components, and integer 3 is the archive number within the class of links with the given crossing size and number of components. For example, (6^3)_1 is the first 3-component link of 6 crossings. This numbering scheme was used by Rolfsen to list all knots up to 10 crossings and all links up to 9 crossings in his book "Knots and Links", Publish & Perish Inc., Berkeley, 1976.

Thistlethwaite Notation

Finally, M. Thistlethwaite has made available via the Knot Atlas his tabulation of all prime alternating links up to 11 crossings. His notation does not refer to the number of components, but rather differentiates between the alternating and non-alternating links. Thus he assigns a number to each alternating link of a given crossing size and independently of these, he assigns a number to each non-alternating link of a given crossing size. Specifically, the format of the Thistlethwaite archive numbers is:

L<integer1>a<integer2> or L<integer1>n<integer2>

so that for example, L6a2 is the second prime alternating link of 6 crossings, while L6n1 is the first (and only) prime non-alternating link of 6 crossings.

Conversion Between Notation

If a gauss code for any knot or link that appears in Rolfsen's tables is entered into Knotilus, the Alexander-Briggs number is given as part of the data of the knot or link. Furthermore, if it is an alternating link that appears in Thistlethwaite's tables, the Thistlethwaite archive number is given as part of the data of the link. Finally, if one has either the Alexander-Briggs number of the Thistlethwaite number of a knot or link, the knot or link may be drawn and its information retrieved by entering the relevant number in the appropriate text boxes near the bottom of the Knotilus web page. Since the archive number from the Knotilus archive of all prime alternating knots and links up to 23 crossings is always presented as part of the data of the knot or link, one may readily convert between the Knotilus archive number, the Alexander-Briggs number, and the Thistlethwaite number (as applicable).