Enumerating the Prime Alternating Links

Abstract

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.

Automatic Molecular Weaving Prototyped by Using Single‐Stranded DNA

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The mean squared writhe of alternating random knot diagrams

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