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.

The construction of DNA nanostructures from branched DNA motifs, or tiles, typically relies on the use of sticky-ended cohesion, owing to the specificity and programmability of DNA sequences. The stability of such constructs when unligated is restricted to a specific range of temperatures, owing to the disruption of base pairing at elevated temperatures. Paranemic (PX) cohesion was developed as an alternative to sticky ends for the cohesion of large topologically closed species that could be purified reliably on denaturing gels. However, PX cohesion is also of limited stability. In this work, we added sticky-ended interactions to PX-cohesive complexes to create interlocked complexes by functionalizing the sticky ends with psoralen, which can form cross-links between the two strands of a double helix. We were able to reinforce the stability of the constructs by creating covalent linkages between the 3'-ends and 5'-ends of the sticky ends; the sticky ends were added to double crossover domains via 3'-3' and 5'-5' linkages. Catenated arrays were obtained either by enzymatic ligation or by UV cross-linking. We have constructed finite-length one-dimensional arrays linked by interlocking loops and have positioned streptavidin-gold particles on these constructs.

Catenation is the process by which cyclic strands are combined like the links of a chain, whereas knotting changes the linking properties of a single strand. In the cell, topoisomerases catalyzing strand passage operations enable the knotting and catenation of DNA so that single- or double-stranded segments can be passed through each other. Here, we use a system of closed DNA structures involving a paranemic motif, called PX-DNA, to bind double strands of DNA together. These PX-cohesive closed molecules contain complementary loops whose linking by Escherichia coli topoisomerase 1 (Topo 1) leads to various types of catenated and knotted structures. We were able to obtain specific DNA topological constructs by varying the lengths of the complementary tracts between the complementary loops. The formation of the structures was analyzed by denaturing gel electrophoresis, and the various topologies of the constructs were characterized using the program Knotilus.

In [5], four knot operators were introduced and used to construct all prime alternating knots of a given crossing size. An efficient implementation of this construction was made possible by the notion of the master array of an alternating knot. The master array and an implementation of the construction appeared in [6]. The basic scheme (as described in [5]) is to apply two of the operators, D and ROT S, to the prime alternating knots of minimal crossing size n − 1, which results in a large set of prime alternating knots of minimal crossing size n, and then the remaining two operators, T and OT S, are applied to these n crossing knots to complete the production of the set of prime alternating knots of minimal crossing size n.In this paper, we show how to obtain all prime alternating links of a given minimal crossing size. More precisely, we shall establish that given any two prime alternating links of minimal crossing size n, there is a finite sequence of T and OT S operations that transforms one of the links into the other. Consequently, one may select any prime alternating link of minimal crossing size n (which is then called the seed link), and repeatedly apply only the operators T and OT S to obtain all prime alternating links of minimal crossing size n from the chosen seed link. The process may be standardized by specifying the seed link to be (in the parlance of [5]) the unique link of n crossings with group number 1, the (n, 2) torus link. IntroductionIn [5], four knot operators were introduced. Of the four, the two called D and ROT S were simply specific instances of the general splicing operation (see Calvo [1] for an extensive discussion of the splicing operation). A form of D was also used by H. de Fraysseix and P. Ossona de Mendez (see [3]) in their work to characterize Gauss codes, and both T and OT S appeared in Conway's seminal paper [2]. These operators were used in [5] to present a method for the construction of all prime alternating knots of a given minimal crossing size. An efficient implementation of this method was presented in [6].When the operators D and ROT S are applied to the prime alternating knots of minimal crossing size n − 1, the result is a large set of prime alternating knots of minimal crossing size n (in the computational work we have done, about 98% of the total number of knots have been constructed by D and ROT S). If the remaining two operators, T and OT S, are then applied to these n crossing knots, the remaining † Partially supported by Wolfgang Struss, and by NSERC Grant R3046A02.

In this paper, we provide a polynomial time (and space), algorithm that determines satisfiability of 3-SAT. The complexity analysis for the algorithm takes into account no efficiency and yet provides a low enough bound, that efficient versions are practical with respect to today's hardware. We accompany this paper with a serial version of the algorithm without non-trivial efficiencies (link: polynomial3sat.org).