Enumerating the Prime Alternating Knots, Part II

Abstract

This is the second of a part series devoted to enumerating prime alternating knots and links. In Part I, we introduced four operators on knots and showed that if these operators are applied to the set of all prime alternating knots of n crossings, the set of all prime alternating knots of n+1 crossings is obtained. In this paper, we explain how to actually implement the operators in an efficient manner. This relies on a complete invariant that we have introduced, called the master array for a prime alternating knot. 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.