Topological chirality and achirality of links

摘要

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 4_1~2, 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 2_1~2, 6_2~2, 8_8~2, 9_(61)~2, 6_2~3, 8_4~3, 8_6~3, and 8_3~4 is proven by the demonstration that all eight non-oriented links can attain rigidly achiral presentations. Furthermore, we conjecture that 9_(61)~2 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 unprecedented among knots or links.