  We will develop various methods, some are of geometric nature and some are of
algebraic nature, to detect the various achiralities of knots and links in
$S^3$. For example, we show 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
we show that all non-trivial links with $\leq9$ crossings are not achiral
except the Borromean rings. A simple procedure for calculating the
$\eta$-function is given in terms of a crossing change formula and its initial
values.
