ENUMERATING THE PRIME ALTERNATING KNOTS, PART I

Abstract

The enumeration of prime knots has a long and storied history, beginning with the work of T. P. Kirkman [9,10], C. N. Little [14], and P. G. Tait [19] in the late 1800's, and continuing through to the present day, with significant progress and related results provided along the way by J. H. Conway [3], K. A. Perko [17, 18], M. B. Thistlethwaite [6, 8, 15, 16, 20], C. H. Dowker [6], J. Hoste [1, 8], J. Calvo [2], W. Menasco [15, 16], W. B. R. Lickorish [12, 13], J. Weeks [8] and many others. Additionally, there have been many efforts to establish bounds on the number of prime knots and links, as described in the works of O. Dasbach and S. Hougardy [4], D. J. A. Welsh [22], C. Ernst and D. W. Sumners [7], and C. Sundberg and M. Thistlethwaite [21] and others. In this paper, we provide a solution to part of the enumeration problem, in that we describe an efficient inductive scheme which uses a total of four operators to generate all prime alternating knots of a given minimal crossing size, and we prove that the procedure does in fact produce them all. The process proceeds in two steps, where in the first step, two of the four operators are applied to the prime alternating knots of minimal crossing size n to produce approximately 98% of the prime alternating knots of minimal crossing size n+1, while in the second step, the remaining two operators are applied to these newly constructed knots, thereby producing the remaining prime alternating knots of crossing size n+1. The process begins with the prime alternating knot of four crossings, the figure eight knot. In the sequel, we provide an actual implementation of our procedure, wherein we spend considerable effort to make the procedure efficient. One very important aspect of the implementation is a new way of encoding a knot. We are able to assign an integer array (called the master array) to a prime alternating knot in such a way that each regular projection, or plane configuration, of the knot can be constructed from the data in the array, and moreover, two knots are equivalent if and only if their master arrays are identical. A fringe benefit of this scheme is a candidate for the so-called ideal configuration of a prime alternating knot. We have used this generation scheme to enumerate the prime alternating knots up to and including those of 19 crossings. The knots up to and including 17 crossings produced by our generation scheme concurred with those found by M. Thistlethwaite, J. Hoste and J. Weeks (see [8]). The current implementation of the algorithms involved in the generation scheme allowed us to produce the 1,769,979 prime alternating knots of 17 crossings on a five node beowulf cluster in approximately 2.3 hours, while the time to produce the prime alternating knots up to and including those of 16 crossings totalled approximately 45 minutes. The prime alternating knots at 18 and 19 crossings were enumerated using the 48 node Compaq ES-40 beowulf cluster at the University of Western Ontario (we also received generous support from Compaq at the SC 99 conference). The cluster was shared with other users and so an accurate estimate of the running time is not available, but the generation of the 8,400,285 knots at 18 crossings was completed in 17 hours, and the generation of the 40,619,385 prime alternating knots at 19 crossings took approximately 72 hours. With the improvements that are described in the sequel, we anticipate that the knots at 19 crossings will be generated in not more than 10 hours on a current Pentium III personal computer equipped with 256 megabytes of main memory.