Knot Theory Work of Ortho Flint and Stuart Rankin

Files of the gauss codes for the distinct minimal diagrams for the prime alternating knots of 13, 14, 15, and 16 crossings.

A table of the numbers of prime alternating links up to and including those of 24 crossings, listed by crossing size and number of components. The data for the links up to 23 crossings was compiled during the summer of 2006, and it took about 20 days of computing time in all. The 24 crossing prime alternating links were enumerated with a distributed version of the enumeration code, taking 40 hours on September 29th and 30th, 2007. The entire collection of links, except for those of 24 crossings, is available online at, and it takes up about 600 gigabytes. In total, there are 98,517,495,461 prime alternating links of crossing size at most 23, and there are 417,377,448,058 prime alternating links of 24 crossings.

A table of the numbers of oriented prime alternating links from 2 - 23 crossings, listed by crossing size and number of components.

Remarks: March 23, 2007. All corrections (see the bug report on January 31, 2007) to the table above have now been made.

Remarks: January 31, 2007. We have just discovered a bug in our code which caused an error in the 20 crossing 3 component link generation (we generated and catalogued two copies of the same link. We have fixed the code error and are now in the process of rerunning the enumeration from that point.

Remarks: December 2, 2003. N.B. The PAKG binaries, and the algorithm itself, are now out-dated. I will leave access to PAKG for the present in case it is of interest to anyone.

For the time being, we are making binaries available of our suite of programs for the generation and manipulation of the prime alternating knots. We will make the source code available after we have completed some refinements that we have scheduled for next summer. Please note that it takes less time to generate the knots than it would be to transfer the database. As we note below, the entire database is available online, but it is not convenient to download the knots from the online access. Please go to the PAKG web page for details and download access.

Remarks: October 15, 2003. We are now able to provide online access to the prime alternating knot catalog via Knotilus is a web interface to the database of prime alternating knots up to 22 crossings, and provides drawings of knots either selected from the database or described by user-entered gauss code. The drawings are provided in Postscript, PDF and JPEG format.

Remarks: September 1, 2003. We have now made substantial improvements to both memory requirements and the amount of work that needs to be done in the enumeration of the prime alternating knots by our algorithms. View a table showing the results of our enumeration of the prime alternating knots up to 22 crossings (23 crossings have now been added). The table also indicates the resources required to generate the various crossing sizes.

Remarks: March 15, 2004. The 23 crossing knots have now been enumerated. It took 228 hours on a Compaq ES-45 with 32gB of ram, of which we used about 13.2gB, and the disk file at 24 bytes per knot is about 604gB in size.

Remarks: June 26, 2003. We have now completed the implementation of the algorithms as described in Papers I and II below, and run them to produce the prime alternating knots of 20 crossings (completed June 26, 2003). With this revised implementation, we are able to compute the prime alternating knots up to and including the 18 crossing knots on a laptop (winxp) with 512mb of memory in about 6 minutes. To compute the 19 crossing knots (which we had first generated in December 1999 on a beowulf cluster taking about 5 days of compute time), required more memory, so we moved to a 2.8ghz Xeon workstation with 3gB of ram running RH 8.1 Linux. On that machine, the construction of the 19 crossing prime alternating knots (starting with a file of the 18 crossing prime alternating knots) took about 25 minutes. With our current implementation, we needed about 16gB of ram (we used lots of ram to gain speed), so we moved to the beowulf cluster called sharcnet (, which had among its machines a Compaq ES-45 with 32gB of ram. On that machine, it took about 2 hours and 45 minutes to construct the complete set of 20 crossing prime alternating knots from the set of 19 crossing prime alternating knots. We are now in the process of recoding to use dramatically less memory (hopefully with not much of an increase in run time) so that we can compute the 21 crossing prime alternating knots on the Xeon workstation.

In the first paper in the list, we introduce four operators on knots and show that, when used according to very simple rules on the prime alternating knots of n crossings, the set of all prime alternating knots of n+1 crossings is obtained. The second paper in the list explains how to actually implement the operators in an efficient manner, although that is in a sense secondary to the appearance of the master array for a prime alternating knot in the second paper. The master array is an ideal knot invariant, in the sense that two prime alternating knots are equivalent if and only if their master arrays are identical. The master array can be used to construct an ideal knot configuration for a prime alternating knot, by which we mean that two prime alternating knots, each in their ideal configuration, are equivalent if and only if they are identical.

By December 1, 1999, we had successfully run an early implementation of the algorithms described in Part II on the UWO Compaq ES-40 48 node beowulf cluster to produce the 40,619,385 prime alternating knots of 19 crossings. At that time, we did not utilize memory very efficiently, and we were not able to continue further, since the number of prime alternating knots is increasing roughly by a factor of 5 with each increase in crossing size at these levels. We have subsequently made very big improvements both in memory utilization and in the amount of work that must be done, and although we have not finished coding the new algorithms, preliminary data suggests that we should be doing approximately 1/6 as much work as before. This would mean that we could compute the 19 crossing knots on a workstation with less than 512mb of ram in less than half a day. The 20 crossing knots would require approximately 2.5 gB of ram and would take approximately 3 days to build. Beyond that, we will probably go on two or three more crossing sizes on the new very large beowulf cluster currently being installed at Western, but at that point, we are starting to use between 100-500 gB of hard disk space for storage (depending on how compact a representation we store), so that might be the end of it for us. The second paper contains full details of the improved algorithms and the criteria for reduction in the amount of computation to be carried out. The third paper in the series establishes a method for enumerating the prime alternating links. It is shown that one may choose any prime alternating link diagram of a given minimal crossing size and by applications of just two operators (namely T and OTS) to the selected seed link, one obtains all prime alternating link diagrams of the desired minimal crossing size.

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  1. Enumerating the Prime Alternating Knots, Part I, joint work by Ortho Flint, John Schermann and Stuart Rankin, (submitted in March of 2001, accepted by JKTR in April 2001). Adobe PDF Format, Adobe Postscript.
  2. Enumerating the Prime Alternating Knots, Part II, joint work by Ortho Flint, John Schermann and Stuart Rankin (submitted in March of 2001, accepted by JKTR in April 2001). Adobe PDF Format, Adobe Postscript.
  3. It presents an implementation of the scheme described in Part I. An early version of this implementation found that at 18 crossings, there were 8,400,285 prime alternating knots, and at 19 crossings, there were 40,619,385 prime alternating knots.

  4. Enumerating the Prime Alternating Links, joint work by Ortho Flint, and Stuart Rankin, (submitted in June of 2002, accepted by JKTR in February 2003). Adobe PDF Format, Adobe Postscript.