| Abstrakt: |
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 ROTS, 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 OTS, 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 OTS 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 OTS 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. [ABSTRACT FROM AUTHOR] |
