Topological chirality and achirality of links

Empirical and analytical methods employed in the detection of topological chirality and achirality (amphicheirality) in oriented and non-oriented links are critically examined.U-polynomials of non-oriented links are modified for use in the detection of topological chirality. By use of this method, all but eight (listed below) non-oriented links with up to four components and nine crossings are proven to be topologically chiral, including 412, the abstract model of the only topologically chiral, non-oriented catenane (chemical link) synthesized so far. The topological chirality of certain 3-Borromean links is similarly proven. The amphicheirality of 212 622 882 9612 632 843 863 and 843 is proven by the demonstration that all eight non-oriented links can attain rigidly achiral presentations. Furthermore, we conjecture that 9612 and a two-component, oriented link with an 11-crossing diagram are the first members of, respectively, a class of non-oriented and a class of oriented amphicheiral, non-alternating, prime links with odd crossing numbers. Amphicheirality combined with an odd crossing number is unprece dented among knots or links.

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