Achirality of knots and links (Q1348240)

An oriented link \(L\) of the three-sphere is achiral (or amphichiral) if there is an orientation-reversing auto-diffeomorphism \(g\) of the three-sphere such that the restriction of \(g\) to the link \(L\) is the identity. By considering the action of \(g\) on the orientation of each component of \(L\), one can define a more general notion of chirality. In the paper under review, various algebraic and geometric methods are developed to detect various achiralities of knots and links in the three-sphere. In particular, the \(\eta\)-functions of Kojima and Yamasaki are applied to the study of chirality of links. The authors provide many necessary conditions for a link to be achiral. The obtained conditions are illustrated by some examples. For example, it is shown that the twisted Whitehead double of a knot is achiral if and only if the double is the unknot or the figure eight knot, and that all non-trivial links with less than nine crossings are chiral except the Borromean rings.