1.11 Noninvertible KLs An orientation of a knot is defined by choosing a direction to travel around the knot. Hence, for every nonoriented knot K, we have two different oriented knots denoted by K¢ and K". A knot K is called invertible if the oriented knots K¢ and K" are equivalent. Otherwise, it is called noninvertible. The existence of noninvertible knots was shown by H.F. Trotter (1963), who discovered the noninvertible knot 7,5,3 and the whole family of noninvertible pretzel knots (2p+1),(2q+1),(2r+1) (p ¹ q, p ¹ r, q ¹ r). Unlike in 1963, today we know that almost all knots are noninvertible (Murasugi, 1996). The number of noninvertible and the number of invertible knots with 3 £ n £ 16 crossings is given in the following table (the sequences A052403 and A052402 from the The OnLine Encyclopedia of Integer Sequences by N. Sloane):
The first noninvertible knot from Rolfsen's knot tables is the knot 8_{17} (.2.2 in the Conway notation). By composing two copies of this knot, one with the matching orientations, and the other with different orientations, we get two distinct composite knots which are not equivalent. A. Kawauchi (1979) proved that there is no deformation of the knot .2.2 that reverses the orientation of the knot. The thirty six noninvertible knots with ten crossings of fewer are identified by R. Hartley (1983). So far, no one has developed a general technique to recognize noninvertible knots. In recognizing achiral knots, we can use their antisymmetrical rigid representations in Â^{3}, or on the sphere S^{3}. Every KL that has an antisymmetrical presentation in Â^{3}, has it on S^{3} as well, but not necessarily vice versa. For example, figureeight knot has both presentations antisymmetrical: its nonminimal diagram is invariant with regard to a rotational antireflection of order 4 (i.e., rotational reflection followed by vertex sign change), and its diagram coming from S^{3} is centroantisymmetrical. Achiral knot 8_{17} has a more remarkable property: it has a centroantisymmetrical presentation coming from S^{3}, but has no antisymmetrical presentation in Â^{3}. Therefore, it is a topological rubber glove in Â^{3} (Flapan, 1998, 2000). The program Snappea by J. Weeks computes knot symmetry group and recognizes noninvertible knots. For checking invertibility of knots you can use also the program Knotscape containing Snappea 2.0 as its part.
